.. _chapter-conventions:

==============================
Conventions for Coding in 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 coding conventions
=========================

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

* http://www.python.org/dev/peps/pep-0008
* http://www.python.org/dev/peps/pep-0257

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.

- 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()

  instead of the CamelCase convention

  ::

      def SetSomeValue()

- Use CamelCase for class names and major functions that create
  objects, e.g. ``PolynomialRing``.

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``.


File and directory names
========================

Python Sage library code 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.

.. 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``.  (For example, see
   ``SAGE_ROOT/devel/sage/sage/ext/notes/``.)


An example is worth a thousand words
====================================

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>
    ...

    AUTHORS:

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

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

    ...

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

    ...

    Lots and lots of examples.
    """

    #*****************************************************************************
    #       Copyright (C) 2010 YOUR NAME <your email>
    #
    #  Distributed under the terms of the GNU General Public License (GPL)
    #  as published by the Free Software Foundation; either version 2 of
    #  the License, or (at your option) any later version. 
    #                  http://www.gnu.org/licenses/ 
    #*****************************************************************************

The following is the top of the file
``SAGE_ROOT/devel/sage/sage/rings/integer.pyx``, which
contains the implementation for :math:`\ZZ`.

::

    r"""
    Elements of the ring `\ZZ` of integers

    AUTHORS:

    - William Stein (2005): initial version

    - Gonzalo Tornaria (2006-03-02): vastly improved python/GMP
      conversion; hashing

    - Didier Deshommes (2006-03-06): numerous examples
      and docstrings

    - William Stein (2006-03-31): changes to reflect GMP bug fixes

    - William Stein (2006-04-14): added GMP factorial method (since it's
      now very fast).

    - David Harvey (2006-09-15): added nth_root, exact_log

    - David Harvey (2006-09-16): attempt to optimise Integer constructor

    - Rishikesh (2007-02-25): changed quo_rem so that the rem is positive

    - David Harvey, Martin Albrecht, Robert Bradshaw (2007-03-01):
      optimized Integer constructor and pool

    - Pablo De Napoli (2007-04-01): multiplicative_order should return
      +infinity for non zero numbers

    - Robert Bradshaw (2007-04-12): is_perfect_power, Jacobi symbol (with
      Kronecker extension).  Convert some methods to use GMP directly
      rather than pari, Integer(), PY_NEW(Integer)

    - David Roe (2007-03-21): sped up valuation and is_square, added
      val_unit, is_power, is_power_of and divide_knowing_divisible_by

    - Robert Bradshaw (2008-03-26): gamma function, multifactorials

    - Robert Bradshaw (2008-10-02): bounded squarefree part

    EXAMPLES:

    Add 2 integers::

        sage: a = Integer(3) ; b = Integer(4)
        sage: a + b == 7
        True

    Add an integer and a real number::

        sage: a + 4.0
        7.00000000000000

    Add an integer and a rational number::

        sage: a + Rational(2)/5
        17/5

    Add an integer and a complex number::

        sage: b = ComplexField().0 + 1.5
        sage: loads((a+b).dumps()) == a+b
        True

        sage: z = 32
        sage: -z
        -32
        sage: z = 0; -z
        0
        sage: z = -0; -z
        0
        sage: z = -1; -z
        1

    Multiplication::

        sage: a = Integer(3) ; b = Integer(4)
        sage: a * b == 12
        True
        sage: loads((a * 4.0).dumps()) == a*b
        True
        sage: a * Rational(2)/5
        6/5

    ::

        sage: list([2,3]) * 4
        [2, 3, 2, 3, 2, 3, 2, 3]

    ::

        sage: 'sage'*Integer(3)
        'sagesagesage'

    COERCIONS:

    Return a version of this integer in the multi-precision floating
    real field R::

        sage: n = 9390823
        sage: RR = RealField(200)
        sage: RR(n)
        9.3908230000000000000000000000000000000000000000000000000000e6


    """
    #*****************************************************************************
    #       Copyright (C) 2004,2006 William Stein <wstein@gmail.com>
    #       Copyright (C) 2006 Gonzalo Tornaria <tornaria@math.utexas.edu>
    #       Copyright (C) 2006 Didier Deshommes <dfdeshom@gmail.com>
    #       Copyright (C) 2007 David Harvey <dmharvey@math.harvard.edu>
    #       Copyright (C) 2007 Martin Albrecht <malb@informatik.uni-bremen.de>
    #       Copyright (C) 2007,2008 Robert Bradshaw <robertwb@math.washington.edu>
    #       Copyright (C) 2007 David Roe <roed314@gmail.com>
    #
    #  Distributed under the terms of the GNU General Public License (GPL)
    #  as published by the Free Software Foundation; either version 2 of
    #  the License, or (at your option) any later version. 
    #                  http://www.gnu.org/licenses/ 
    #*****************************************************************************

All code included with Sage must be licensed under the GPLv2+ or a
less restrictive license (e.g. the BSD license). It is very important
that you include your name in the AUTHORS log so that everybody who
submits code to Sage receives proper credit [2]_. If ever you feel you
are not receiving proper credit for anything you submit to Sage,
please let the development team know!


.. _section-docstrings:

Documentation strings
=====================


Docstring markup with ReST and Sphinx
-------------------------------------

**Every** function must have a docstring that includes the following
information. Source files in the Sage library contain numerous
examples on how to format your documentation, so you could use them as
a guide.

-  A one-sentence description of the function, followed by a blank line
   and ending in a period. It prescribes the function or method's
   effect as a command ("Do this", "Return that"), not as a
   description; e.g. don't write "Returns the pathname ..."

-  An INPUT and an OUTPUT block for input and output arguments (see
   below for format). 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; you do not have
   to put a precise type name such as ``int``. The INPUT block
   describes the expected input to your function or method, while the
   OUTPUT block describes the expected output of the
   function/method. If appropriate, you need to describe any default
   values for the input arguments. For example::

       INPUT:

       - ``p`` -- (default: 2) a positive prime integer.

       OUTPUT:

       A 5-tuple consisting of integers in this order:

       1. the smallest primitive root modulo p
       2. the smallest prime primitive root modulo p
       3. the largest primitive root modulo p
       4. the largest prime primitive root modulo p
       5. total number of prime primitive roots modulo p

   Some people prefer to format their OUTPUT section as a block by
   using a dash. That is acceptable as well::

       OUTPUT:

       - The plaintext resulting from decrypting the ciphertext ``C``
         using the Blum-Goldwasser decryption algorithm.

-  Instead of INPUT and OUTPUT blocks, you can include descriptions of
   the arguments and output using Sphinx/ReST markup, as described in
   http://sphinx.pocoo.org/markup/desc.html#info-field-lists.  See
   below for an example.

-  An EXAMPLES block for examples. This is not optional. These
   examples are used for automatic testing before each release and new
   functions without these doctests will not be accepted for inclusion
   with Sage.

-  A SEEALSO block (optional) with links to related things in Sage. A SEEALSO
   block should start with ``.. SEEALSO::``. It can also be the lower-case form
   ``.. seealso::``. However, you are encouraged to use the upper-case form
   ``.. SEEALSO::``. See :ref:`chapter-sage_manuals_links` for details on how
   to setup link in Sage.  Here's an example of a SEEALSO block::

       .. SEEALSO::

           :ref:`chapter-sage_manuals_links`

-  An ALGORITHM block (optional) which indicates what software
   and/or what algorithm is used. For example
   ``ALGORITHM: Uses Pari``. Here's a longer example that describes an
   algorithm used. Note that it also cites the reference where this
   algorithm can be found::

       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`.

       REFERENCES:

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

   You can also number the steps in your algorithm using the hash-dot
   symbol. This way, the actual numbering of the steps are
   automatically taken care of when you build the documentation::

        ALGORITHM:

        The Blum-Goldwasser decryption algorithm is described in Algorithm
        8.56, page 309 of [MenezesEtAl1996]_. The algorithm works as follows:

        #. Let `C` be the ciphertext `C = (c_1, c_2, \dots, c_t, x_{t+1})`.
           Then `t` is the number of ciphertext sub-blocks and `h` is the
           length of each binary string sub-block `c_i`.
        #. Let `(p, q, a, b)` be the private key whose corresponding
           public key is `n = pq`. Note that `\gcd(p, q) = ap + bq = 1`.
        #. Compute `d_1 = ((p + 1) / 4)^{t+1} \bmod{(p - 1)}`.
        #. Compute `d_2 = ((q + 1) / 4)^{t+1} \bmod{(q - 1)}`.
        #. Let `u = x_{t+1}^{d_1} \bmod p`.
        #. Let `v = x_{t+1}^{d_2} \bmod q`.
        #. Compute `x_0 = vap + ubq \bmod n`.
        #. For `i` from 1 to `t`, do:

           #. Compute `x_i = x_{t-1}^2 \bmod n`.
           #. Let `p_i` be the `h` least significant bits of `x_i`.
           #. Compute `m_i = p_i \oplus c_i`.

        #. The plaintext is `m = m_1 m_2 \cdots m_t`.

-  A NOTE block for special notes (optional). Include information
   such as purpose etc. A NOTE block should start with
   ``.. NOTE::``. You can also use the lower-case version
   ``.. note::``, but do not mix lower-case with upper-case. However,
   you are encouraged to use the upper-case version ``.. NOTE::``. If
   you want to put anything within the NOTES block, you should
   indent it at least 4 spaces (no tabs). Here's an example of a NOTE
   block::

       .. NOTE::

           You should note that this sentence is indented at least 4
           spaces. Avoid tab characters as much as possible when
           writing code or editing the Sage documentation. You should
           follow Python conventions by using spaces only.

- A WARNING block for critical information about your code. For
  example, the WARNING block might include information about when or
  under which conditions your code might break, or information that
  the user should be particularly aware of. A WARNING block should start
  with ``.. WARNING::``. It can also be the lower-case form
  ``.. warning::``. However, you are encouraged to use the upper-case
  form ``.. WARNING::``. Here's an example of a WARNING block::

      .. 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 room for improvements. The TODO block might
  contains disabled doctests to demonstrate the desired feature.  A TODO block
  should start with ``.. TODO::``. It can also be the lower-case form
  ``.. todo::``. However, you are encouraged to use the upper-case form
  ``.. TODO::``. Here's an example of a TODO block::

      .. TODO::

          Improve further function ``have_fresh_beers`` using algorithm
          ``buy_a_better_fridge``::

              sage: have_fresh_beers('Bière de l\'Yvette') # todo: not implemented
              Enjoy !

- A REFERENCES block to list books or papers (optional). This block serves
  a similar purpose to a list of references in a research paper, or a
  bibliography in a monograph. If your method, function or class uses an
  algorithm that can be found in a standard reference, you should list
  that reference under this block. The Sphinx/ReST markup for
  citations is described at
  http://sphinx.pocoo.org/rest.html#citations. See below for an example.
  Sage also add specific markup for links to sage trac tickets and
  Wikipedia. See :ref:`chapter-sage_manuals_links`. Here's an example of a
  REFERENCES block::

    This docstring is referencing [SC]_. Just remember that references
    are global, so we can also reference to [Nat2000]_ in the ALGORITHM
    block, even if it is in a separate file. However we would not
    include the reference here since it would cause a conflict.

    REFERENCES:

    .. [SC] Conventions for coding in sage.
       http://www.sagemath.org/doc/developer/conventions.html.

-  An AUTHORS block (optional, but encouraged for important
   functions, so users can see from the docstring who wrote it and
   therefore whom to contact if they have questions).

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

.. skip

::

    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 ...

        OUTPUT:

        The point as a tuple.

        .. SEEALSO::

            :func:`line`

        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 x (=r) to an integer

        .. NOTE::

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

        ...

        REFERENCES:

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

        AUTHORS:

        - William Stein (2005-01-03)

        - First_name Last_name (yyyy-mm-dd)
        """
        <body of the function>

If you used Sphinx/ReST markup for the arguments, the beginning of the
docstring would look like this::

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

        :param x: the description of the argument x goes here.
        If it contains multiple lines, all the lines after the
        first need to be indented.

        :type x: integer; default 1

        :param y: the ...

        :type y: integer; default 2

        :returns: the ...

        :rtype: integer, the return type

You are strongly encouraged to:

-  Use nice LaTeX formatting everywhere. If you use backslashes,
   either use double backslashes or place an "r" right before the
   first triple opening quote. For example,

   ::

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

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

   You can also use the MATH block to format complicated mathematical
   expressions::

       .. MATH::

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

   Note that the MATH block is automatically wrapped in a latex math
   environment (i.e. in ``\[ \]`` or ``$$``, etc.). To use aligned equations,
   use the **aligned** environment::

       .. MATH::

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

   If you wish to explicitly not wrap the MATH block, make the first line of
   the indented block ``:nowrap:``::

       .. MATH::
           :nowrap:

           This is now plain text so I can do things like $x = 5$.

   .. WARNING::

       With or without ``:nowrap:``, the *html* documentation output
       currently will work if you use environments such as **align**
       which wrap their contents in math mode. However, ``:nowrap:``
       is necessary for the *pdf* documentation to build correctly.

   .. NOTE::

      In ReST documentation, you use backticks \` to mark LaTeX code
      to be typeset.  In Sage docstrings, unofficially you may use
      dollar signs instead -- "unofficially" means that it ought to
      work, but might be a little buggy.  Thus ```x^2 + y^2 = 1``` and
      ``$x^2 + y^2 = 1$`` should produce identical output, typeset in math
      mode.

      LaTeX style: typeset standard rings and fields like the integers
      and the real numbers using the locally-defined macro ``\\Bold``,
      as in ``\\Bold{Z}`` for the integers. This macro is defined to be
      ordinary bold-face ``\\mathbf`` by default, but users can switch to
      blackboard-bold ``\\mathbb`` and back on-the-fly by using
      ``latex.blackboard_bold(True)`` and
      ``latex.blackboard_bold(False)``.

      The docstring will be available interactively (for the "def
      point..." example above, by typing "point?" at the "sage:"
      prompt) and also in the reference manual. When viewed
      interactively, LaTeX code has the backslashes stripped from it,
      so "\\cos" will appear as "cos".

      Because of the dual role of the docstring, you need to strike a
      balance between readability (for interactive help) and using
      perfect LaTeX code (for the reference manual).  For instance,
      instead of using "\\frac{a}{b}", use "a/b" or maybe "a b^{-1}".
      Also keep in mind that some users of Sage are not familiar with
      LaTeX; this is another reason to avoid complicated LaTeX
      expressions in docstrings, if at all possible: "\\frac{a}{b}"
      will be obscure to someone who doesn't know any LaTeX.

      Finally, a few non-standard LaTeX macros are available to help
      achieve this balance, including "\\ZZ", "\\RR", "\\CC", and
      "\\QQ".  These are names of Sage rings, and they are typeset
      using a single boldface character; they allow the use of "\\ZZ"
      in a docstring, for example, which will appear interactively as
      "ZZ" while being typeset as "\\Bold{Z}" in the reference
      manual.  Other examples are "\\GF" and "\\Zmod", each of which
      takes an argument: "\\GF{q}" is typeset as "\\Bold{F}_{q}" and
      "\\Zmod{n}" is typeset as "\\Bold{Z}/n\\Bold{Z}".  See the
      file ``$SAGE_ROOT/devel/sage/sage/misc/latex_macros.py`` for a
      full list and for details about how to add more macros.

-  Liberally describe what the examples do. Note that there must be
   a blank line after the example code and before the explanatory text
   for the next example (indentation is not enough).

-  Illustrate any exceptions raised by the function with examples,
   as given above. (It is an error to ...; In particular, use ...)

-  Include many examples. These are automatically tested on a regular
   basis, 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. Note that new
   functions without doctests will not be accepted for inclusion in Sage.

.. WARNING::

    Functions whose names start with an underscore do not currently
    appear in the reference manual, so avoid putting crucial
    documentation in their docstrings. In particular, if you are
    defining a class, you might put a long informative docstring after
    the class definition, not for the ``__init__`` method. For example,
    from the file ``SAGE_ROOT/devel/sage/sage/crypto/classical.py``::

        class HillCryptosystem(SymmetricKeyCryptosystem):
            """
            Create a Hill cryptosystem defined by the `m` x `m` matrix space
            over `\mathbf{Z} / N \mathbf{Z}`, where `N` is the alphabet size of
            the string monoid ``S``.

            INPUT:

            - ``S`` -- a string monoid over some alphabet

            - ``m`` -- a positive integer; the block length of matrices that
              specify block permutations

            OUTPUT:

            - A Hill cryptosystem of block length ``m`` over the alphabet ``S``.

            EXAMPLES::

                sage: S = AlphabeticStrings()
                sage: E = HillCryptosystem(S,3)
                sage: E
                Hill cryptosystem on Free alphabetic string monoid on A-Z of block length 3
        """

    and so on, while the ``__init__`` method starts like this::

        def __init__(self, S, m):
            """
            See ``HillCryptosystem`` for full documentation.

            EXAMPLES::

                ...
            """

    Note also that the first docstring is printed if users type
    "HillCryptosystem?" at the "sage:" prompt.

    (Before Sage 3.4, the reference manual used to include methods
    starting with underscores, so you will probably find many examples
    in the code which don't follow this advice...)


Automatic testing
-----------------

The code in the examples should pass automatic testing. This means
that if the above code is in the file ``f.py`` (or ``f.sage``), then
``sage -t f.py`` should not give any error messages. Testing occurs
with full Sage preparsing of input within the standard Sage shell
environment, as described in :ref:`section-preparsing`. **Important:**
The file ``f.py`` is not imported when running tests unless you have
arranged that it be imported into your Sage environment, i.e. unless
its functions are available when you start Sage using the ``sage``
command. For example, the function ``AA()`` in the file
``SAGE_ROOT/devel/sage/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.


.. _section-further_conventions:

Further conventions for automated testing of examples
-----------------------------------------------------

The Python script ``SAGE_LOCAL/bin/sage-runtests`` implements
documentation testing in Sage (see :ref:`chapter-testing` for more
details). When writing documentation, keep the following points in
mind:

-  All input is preparsed before being passed to Python, e.g. ``2/3``
   is replaced by ``Integer(2)/Integer(3)``, which evaluates to
   ``2/3`` as a rational instead of the Python int ``0``. For more
   information on preparsing, see :ref:`section-preparsing`.

-  If a test outputs to a file, the file should be a temporary file.
   Use :func:`tmp_filename` to get a temporary filename,
   or :func:`tmp_dir` to get a temporary directory.
   For example (taken from the file
   ``SAGE_ROOT/devel/sage/sage/plot/graphics.py``)::

       sage: plot(x^2 - 5, (x, 0, 5), ymin=0).save(tmp_filename(ext='.png'))

-  If a test line contains the text ``random``, it is executed by
   ``sage-runtests`` but ``sage-runtests`` does not check that the
   output agrees with the output in the documentation string. For
   example, the docstring for the ``__hash__`` method for
   ``CombinatorialObject`` in
   ``SAGE_ROOT/devel/sage/sage/combinat/combinat.py`` includes
   the lines

   .. skip

   ::

           sage: hash(c) #random
           1335416675971793195
           sage: c._hash #random
           1335416675971793195

   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. See :ref:`chapter-randomtesting` for details on this
   framework.

-  If a line contains the text ``long time`` then that line is not
   tested unless the ``--long`` option is given, e.g.
   ``sage -t --long f.py``. Use this to include examples that take more
   than about a second to run. These will not be run regularly during
   Sage development, but will get run before major releases. No
   example should take more than about 30 seconds.

   For instance, here is part of the docstring from the ``regulator``
   method for rational elliptic curves, from the file
   ``SAGE_ROOT/devel/sage/sage/schemes/elliptic_curves/ell_rational.py``:

   ::

       sage: E = EllipticCurve([0, 0, 1, -1, 0])
       sage: E.regulator()              # long time (1 second)
       0.0511114082399688

-  If a line contains ``tol`` or ``tolerance``, numerical results are only
   verified to the given tolerance. This may be prefixed by ``abs[olute]``
   or ``rel[ative]`` to specify whether to measure absolute or relative
   error; this defaults to relative error except when the expected value 
   is exactly zero:

   ::

       sage: RDF(pi)                               # abs tol 1e-5
       3.14159
       sage: [10^n for n in [0.0 .. 4]]            # rel tol 2e-4
       [0.9999, 10.001, 100.01, 999.9, 10001]

   This can be useful when the exact output is subject to rounding error
   and/or processor floating point arithmetic variation.  Here are some 
   more examples.
   
   A singular value decomposition of a matrix will produce two unitary
   matrices.  Over the reals, this means the inverse of the matrix is
   equal to its transpose.  We test this result by applying the norm to
   a matrix difference.  The result will usually be a "small" number,
   distinct from zero.
   
   ::

       sage: A = matrix(RDF, 8, range(64))
       sage: U, S, V = A.SVD()
       sage: (U.transpose()*U-identity_matrix(8)).norm(p=2)    # abs tol 1e-10
       0.0

   The 8-th cyclotomic field is generated by the complex number
   `e^\frac{i\pi}{4}`.  Here we compute a numerical approximation::

       sage: K.<zeta8> = CyclotomicField(8)
       sage: N(zeta8)                             # absolute tolerance 1e-10
       0.7071067812 + 0.7071067812*I

   A relative tolerance on a root of a polynomial.  Notice that the
   root should normally print as ``1e+16``, or something similar.
   However, the tolerance testing causes the doctest framework to use
   the output in a *computation*, so other valid text representations
   of the predicted value may be used.  However, they must fit the
   pattern defined by the regular expression ``float_regex`` in
   :mod:`sage.doctest.parsing`.

   ::

       sage: y = polygen(RDF, 'y')
       sage: p = (y - 10^16)*(y-10^(-13))*(y-2); p
       y^3 - 1e+16*y^2 + 2e+16*y - 2000.0
       sage: p.roots(multiplicities=False)[2]     # relative tol 1e-10
       10000000000000000

-  If a line contains ``not implemented`` or ``not tested``, it is
   never tested. It is good to include lines like this to make clear
   what we want Sage to eventually implement::

       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.

- If one of the first 10 lines of a file starts with ``r""" nodoctest``
  (or ``""" nodoctest`` or ``# nodoctest`` or ``% nodoctest`` or
  ``.. nodoctest``, or any of these with different spacing), then
  that file will be skipped.
  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.

- If a line contains ``# optional - PKGNAME`` (where the ``#`` may be any
  non-letter non-space character), it is not tested unless
  the ``--optional=PKGNAME`` flag is passed to ``sage -t``.
  Mark a doctest as ``optional`` if it requires optional packages.
  Running ``sage -t --optional=all f.py`` executes all doctests,
  including all optional tests.
  Running ``sage -t --optional=sage,sloane_database f.py`` runs the
  normal tests (because of ``--optional=sage``),
  as well as those marked as ``# optional - sloane_database``.
  For example, the file
  ``SAGE_ROOT/devel/sage/sage/databases/sloane.py`` contains the lines

  ::

       sage: sloane_sequence(60843)       # optional - internet 

  and ::

       sage: SloaneEncyclopedia[60843]    # optional - sloane_database

  The first of these just needs internet access, while the second
  requires that the "sloane_database" package be installed.  Calling
  ``sage -t --optional=all`` on this file runs both of these tests, while
  calling ``sage -t --optional=sage,internet`` on it will only run the
  first test.
  A test requiring several packages should be marked
  ``# optional - pkg1 pkg2`` and executed by
  ``sage -t --optional=sage,pkg1,pkg2 f.py``.

  .. 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``.

- If you are documenting a known bug in Sage, mark it as ``known bug``
  or ``optional: bug``.  For example::

     The following should yield 4.  See :trac:`2`. ::

        sage: 2+2 # optional: bug
        5

  Then the doctest will be skipped by default, but could be revealed
  by running ``sage -t --optional=sage,bug ...``.  (A doctest marked
  as ``known bug`` gets automatically converted to ``optional bug``).

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.

- Some tests (hashing for example) behave differently on 32-bit and
  64-bit platforms.  You can mark a line (generally the output) with
  either ``# 32-bit`` or ``# 64-bit`` and the testing framework will
  remove any lines that don't match the current architecture.  For
  example::

      sage: z = 32
      sage: z.powermodm_ui(2^32-1, 14)
      Traceback (most recent call last):                              # 32-bit
      ...                                                             # 32-bit
      OverflowError: exp (=4294967295) must be <= 4294967294          # 32-bit
      8              # 64-bit

- You may write tests that span multiple lines.  The best way to do so
  is to use the line continuation marker ``....:``. ::

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

  If you have a long line of code, you may want to consider adding a
  backslash to the end of the line, which tells the doctesting
  framework to join that current line with the next.  This syntax is
  non-standard so may be removed in a future version of Sage, but in
  the mean time it can be useful for breaking up large integers across
  multiple lines::

      sage: n = 123456789123456789123456789\
      ....:     123456789123456789123456789
      sage: n.is_prime()
      False

.. _chapter-testing:

Automated testing
=================

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 :ref:`section-docstrings` for a discussion on how to
include examples in documentation strings and what conventions to
follow. The chapter :ref:`chapter-doctesting` contains a tutorial on
doctesting modules in the Sage library.


.. _section-testpython:

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 :ref:`chapter-doctesting`.

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::

    ::

            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/devel/sage/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/devel/sage/doc/en/tutorial/`` for many
examples of how to include automated testing in ReST documentation
for Sage.

.. _chapter-picklejar:

The pickle jar
==============

Sage maintains a pickle jar at
``SAGE_ROOT/data/extcode/pickle_jar/pickle_jar.tar.bz2`` which is a tar file of
"standard" pickles created by ``sage``. This pickle jar is used to ensure that
sage maintains backward compatibility by have having
:func:`sage.structure.sage_object.unpickle_all` check that ``sage`` can always
unpickle all of the pickles in the pickle jar as part of the standard doc
testing framework.

Most people first become aware of the pickle_jar when their patch breaks the
unpickling of one of the "standard" pickles in the pickle jar due to the
failure of the doctest::

    sage -t devel/sage-main/sage/structure/sage_object.pyx

When this happens an error message is printed which contains the following
hints for fixing the uneatable pickle::

    ----------------------------------------------------------------------
    ** This error is probably due to an old pickle failing to unpickle.
    ** See sage.structure.sage_object.register_unpickle_override for
    ** how to override the default unpickling methods for (old) pickles.
    ** NOTE: pickles should never be removed from the pickle_jar!
    ----------------------------------------------------------------------

For more details about how to fix unpickling errors in the pickle jar
see :func:`sage.structure.sage_object.register_unpickle_override`

.. NOTE::

    Every so often the standard pickle jar should be updated by running the
    doctest suite with the environment variable ``SAGE_PICKLE_JAR`` set, then
    copying the files from ``SAGE_ROOT/tmp/pickle_jar*`` into the standard pickle
    jar.

.. WARNING::

    Sage's pickle jar helps to ensure backward compatibility in sage. Pickles should
    **only** be removed from the pickle jar after the corresponding objects
    have been properly deprecated. Any proposal to remove pickles from the
    pickle jar should first be discussed on sage-devel.

.. _chapter-randomtesting:

Randomized testing
==================

In addition to all the examples in your docstrings, which serve as
both demonstrations and tests of your code, you should consider
creating a test suite. Think of this as a program that will run for a
while and "tries" to crash your code using randomly generated
input. Your test code should define a class ``Test`` with a
``random()`` method that runs random tests. These are all assembled
together later, and each test is run for a certain amount of time on a
regular basis.

For example, see the file
``SAGE_ROOT/devel/sage/sage/modular/modsym/tests.py``.

.. [2]  See http://www.sagemath.org/development-map.html

GlobalOptions
=============

Global options for classes can be defined in Sage using
:class:`~sage.structure.global_options.GlobalOptions`.