Coding in Python for Sage - Developer Guide

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# Coding in Python for Sage

This chapter discusses some issues with, and advice for, coding in Sage.

## Python language standard

Sage follows the time window-based support policy SPEC 0 — Minimum Supported Dependencies for Python versions. The current minimum supported Python version can be found in the `pyproject.toml` file. Accordingly, only language and library features available in this version can be used. If a feature is deprecated in a newer supported version, it must be ensured that deprecation warnings issued by Python do not lead to failures in doctests.

Some key language and library features have been backported to older Python versions using one of two mechanisms:

- `from __future__ import annotations` (see Python reference for __future__) modernizes type annotations according to PEP 563 (Postponed evaluation of annotations). All Sage library code that uses type annotations should include this `__future__` import and follow PEP 563.

- The typing_extensions package is used to backport features from newer versions of the `typing` module. The Sage library declares this package as a dependency.

## Design

If you are planning to develop some new code for Sage, design is important. So think about what your program will do and how that fits into the structure of Sage. In particular, much of Sage is implemented in the object-oriented language Python, and there is a hierarchy of classes that organize code and functionality. For example, if you implement elements of a ring, your class should derive from `sage.structure.element.RingElement`, rather than starting from scratch. Try to figure out how your code should fit in with other Sage code, and design it accordingly.

## Special Sage functions

Functions with leading and trailing double underscores `__XXX__` are all predefined by Python. Functions with leading and trailing single underscores `_XXX_` are defined for Sage. Functions with a single leading underscore are meant to be semi-private, and those with a double leading underscore are considered really private. Users can create functions with leading and trailing underscores.

Just as Python has many standard special methods for objects, Sage also has special methods. They are typically of the form `_XXX_`. In a few cases, the trailing underscore is not included, but this will eventually be changed so that the trailing underscore is always included. This section describes these special methods.

All objects in Sage should derive from the Cython extension class `SageObject`:

    from sage.structure.sage_object import SageObject

    class MyClass(SageObject,...):
        ...

or from some other already existing Sage class:

    from sage.structure.parent import Parent

    class MyFavoriteAlgebra(Parent):
        ...

You should implement the `_latex_` and `_repr_` method for every object. The other methods depend on the nature of the object.

## LaTeX representation

Every object `x` in Sage should support the command `latex(x)`, so that any Sage object can be easily and accurately displayed via LaTeX. Here is how to make a class (and therefore its instances) support the command `latex`.

1. Define a method `_latex_(self)` that returns a LaTeX representation of your object. It should be something that can be typeset correctly within math mode. Do not include opening and closing $’s.

2. Often objects are built up out of other Sage objects, and these components should be typeset using the `latex` function. For example, if `c` is a coefficient of your object, and you want to typeset `c` using LaTeX, use `latex(c)` instead of `c._latex_()`, since `c` might not have a `_latex_` method, and `latex(c)` knows how to deal with this.

3. Do not forget to include a docstring and an example that illustrates LaTeX generation for your object.

4. You can use any macros included in `amsmath`, `amssymb`, or `amsfonts`, or the ones defined in `sage.misc.latex_macros`.

An example template for a `_latex_` method follows. Note that the `.. skip` line should not be included in your code; it is here to prevent doctests from running on this fake example.

    class X:
       ...
       def _latex_(self):
           r"""
           Return the LaTeX representation of X.

           EXAMPLES::

               sage: a = X(1,2)
               sage: latex(a)
               '\\frac{1}{2}'
           """
           return '\\frac{%s}{%s}'%(latex(self.numer), latex(self.denom))

As shown in the example, `latex(a)` will produce LaTeX code representing the object `a`. Calling `view(a)` will display the typeset version of this.

## Print representation

The standard Python printing method is `__repr__(self)`. In Sage, that is for objects that derive from `SageObject` (which is everything in Sage), instead define `_repr_(self)`. This is preferable because if you only define `_repr_(self)` and not `__repr__(self)`, then users can rename your object to print however they like. Also, some objects should print differently depending on the context.

Here is an example of the `_latex_` and `_repr_` functions for the `Pi` class. It is from the file SAGE_ROOT/src/sage/symbolic/constants.py:

    class Pi(Constant):
        """
        The ratio of a circle's circumference to its diameter.

        EXAMPLES::

            sage: pi
            pi
            sage: float(pi) # rel tol 1e-10
            3.1415926535897931
        """
        ...
        def _repr_(self):
            return "pi"

        def _latex_(self):
            return "\\pi"

## Matrix or vector from object

Provide a `_matrix_` method for an object that can be coerced to a matrix over a ring \(R\). Then the Sage function `matrix` will work for this object.

The following is from SAGE_ROOT/src/sage/graphs/generic_graph.py:

    class GenericGraph(SageObject):
        ...
        def _matrix_(self, R=None):
            if R is None:
                return self.am()
            else:
                return self.am().change_ring(R)

        def adjacency_matrix(self, sparse=None, boundary_first=False):
            ...

Similarly, provide a `_vector_` method for an object that can be coerced to a vector over a ring \(R\). Then the Sage function `vector` will work for this object. The following is from the file SAGE_ROOT/src/sage/modules/free_module_element.pyx:

    cdef class FreeModuleElement(element_Vector):   # abstract base class
        ...
        def _vector_(self, R):
            return self.change_ring(R)

## Sage preparsing

To make Python even more usable interactively, there are a number of tweaks to the syntax made when you use Sage from the commandline or via the notebook (but not for Python code in the Sage library). Technically, this is implemented by a `preparse()` function that rewrites the input string. Most notably, the following replacements are made:

- Sage supports a special syntax for generating rings or, more generally, parents with named generators:

  Sage

      sage: R.<x,y> = QQ[]
      sage: preparse('R.<x,y> = QQ[]')
      "R = QQ['x, y']; (x, y,) = R._first_ngens(2)"

  Python

      >>> from sage.all import *
      >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
      >>> preparse('R.<x,y> = QQ[]')
      "R = QQ['x, y']; (x, y,) = R._first_ngens(2)"

- Integer and real literals are Sage integers and Sage floating point numbers. For example, in pure Python these would be an attribute error:

  Sage

      sage: 16.sqrt()
      4
      sage: 87.factor()
      3 * 29

  Python

      >>> from sage.all import *
      >>> Integer(16).sqrt()
      4
      >>> Integer(87).factor()
      3 * 29

- Raw literals are not preparsed, which can be useful from an efficiency point of view. In Sage raw integer and floating literals are followed by an “r” (or “R”) for raw, meaning not preparsed. For example:

  Sage

      sage: a = 393939r
      sage: a
      393939
      sage: type(a)
      <... 'int'>
      sage: b = 393939
      sage: type(b)
      <class 'sage.rings.integer.Integer'>
      sage: a == b
      True

  Python

      >>> from sage.all import *
      >>> a = 393939
      >>> a
      393939
      >>> type(a)
      <... 'int'>
      >>> b = Integer(393939)
      >>> type(b)
      <class 'sage.rings.integer.Integer'>
      >>> a == b
      True

- Raw literals can be very useful in certain cases. For instance, Python integers can be more efficient than Sage integers when they are very small. Large Sage integers are much more efficient than Python integers since they are implemented using the GMP C library.

Consult the file `preparser.py` for more details about Sage preparsing, more examples involving raw literals, etc.

When a file `foo.sage` is loaded or attached in a Sage session, a preparsed version of `foo.sage` is created with the name `foo.sage.py`. The beginning of the preparsed file states:

    This file was *autogenerated* from the file foo.sage.

You can explicitly preparse a file with the `--preparse` command-line option: running

    $ sage --preparse foo.sage

creates the file `foo.sage.py`.

The following files are relevant to preparsing in Sage:

1. SAGE_ROOT/src/bin/sage

2. SAGE_ROOT/src/bin/sage-preparse

3. SAGE_ROOT/src/sage/repl/preparse.py

In particular, the file `preparse.py` contains the Sage preparser code.

## The Sage coercion model

The primary goal of coercion is to be able to transparently do arithmetic, comparisons, etc. between elements of distinct sets. For example, when one writes \(3 + 1/2\), one wants to perform arithmetic on the operands as rational numbers, despite the left term being an integer. This makes sense given the obvious and natural inclusion of the integers into the rational numbers. The goal of the coercion system is to facilitate this (and more complicated arithmetic) without having to explicitly map everything over into the same domain, and at the same time being strict enough to not resolve ambiguity or accept nonsense.

The coercion model for Sage is described in detail, with examples, in the Coercion section of the Sage Reference Manual.

## Mutability

Parent structures (e.g. rings, fields, matrix spaces, etc.) should be immutable and globally unique whenever possible. Immutability means, among other things, that properties like generator labels and default coercion precision cannot be changed.

Global uniqueness while not wasting memory is best implemented using the standard Python weakref module, a factory function, and module scope variable.

Certain objects, e.g. matrices, may start out mutable and become immutable later. See the file SAGE_ROOT/src/sage/structure/mutability.py.

## The __hash__ special method

Here is the definition of `__hash__` from the Python reference manual:

> Called by built-in function `hash()` and for operations on members of hashed collections including `set`, `frozenset`, and `dict`. `__hash__()` should return an integer. The only required property is that objects which compare equal have the same hash value; it is advised to mix together the hash values of the components of the object that also play a part in comparison of objects by packing them into a tuple and hashing the tuple.

> If a class does not define an `__eq__()` method it should not define a `__hash__()` operation either; if it defines `__eq__()` but not `__hash__()`, its instances will not be usable as items in hashable collections. If a class defines mutable objects and implements an `__eq__()` method, it should not implement `__hash__()`, since the implementation of hashable collections requires that a key’s hash value is immutable (if the object’s hash value changes, it will be in the wrong hash bucket).

See https://docs.python.org/3/reference/datamodel.html#object.__hash__ for more information on the subject.

Notice the phrase, “The only required property is that objects which compare equal have the same hash value.” This is an assumption made by the Python language, which in Sage we simply cannot make (!), and violating it has consequences. Fortunately, the consequences are pretty clearly defined and reasonably easy to understand, so if you know about them they do not cause you trouble. The following example illustrates them pretty well:

Sage

    sage: v = [Mod(2,7)]
    sage: 9 in v
    True
    sage: v = set([Mod(2,7)])
    sage: 9 in v
    False
    sage: 2 in v
    True
    sage: w = {Mod(2,7):'a'}
    sage: w[2]
    'a'
    sage: w[9]
    Traceback (most recent call last):
    ...
    KeyError: 9

Python

    >>> from sage.all import *
    >>> v = [Mod(Integer(2),Integer(7))]
    >>> Integer(9) in v
    True
    >>> v = set([Mod(Integer(2),Integer(7))])
    >>> Integer(9) in v
    False
    >>> Integer(2) in v
    True
    >>> w = {Mod(Integer(2),Integer(7)):'a'}
    >>> w[Integer(2)]
    'a'
    >>> w[Integer(9)]
    Traceback (most recent call last):
    ...
    KeyError: 9

Here is another example:

Sage

    sage: R = RealField(10000)
    sage: a = R(1) + R(10)^-100
    sage: a == RDF(1)  # because the a gets coerced down to RDF
    True

[Python]

    >>> from sage.all import *
    >>> R = RealField(Integer(10000))
    >>> a = R(Integer(1)) + R(Integer(10))**-Integer(100)
    >>> a == RDF(Integer(1))  # because the a gets coerced down to RDF
    True

but `hash(a)` should not equal `hash(1)`.

Unfortunately, in Sage we simply cannot require

    (#)   "a == b ==> hash(a) == hash(b)"

because serious mathematics is simply too complicated for this rule. For example, the equalities `z == Mod(z, 2)` and `z == Mod(z, 3)` would force `hash()` to be constant on the integers.

The only way we could “fix” this problem for good would be to abandon using the `==` operator for “Sage equality”, and implement Sage equality as a new method attached to each object. Then we could follow Python rules for `==` and our rules for everything else, and all Sage code would become completely unreadable (and for that matter unwritable). So we just have to live with it.

So what is done in Sage is to attempt to satisfy `(#)` when it is reasonably easy to do so, but use judgment and not go overboard. For example,

Sage

    sage: hash(Mod(2,7))
    2

Python

    >>> from sage.all import *
    >>> hash(Mod(Integer(2),Integer(7)))
    2

The output 2 is better than some random hash that also involves the moduli, but it is of course not right from the Python point of view, since `9 == Mod(2,7)`. The goal is to make a hash function that is fast, but within reason respects any obvious natural inclusions and coercions.

## Exceptions

Please avoid catch-all code like this:

    try:
        some_code()
    except:               # bad
        more_code()

If you do not have any exceptions explicitly listed (as a tuple), your code will catch absolutely anything, including `ctrl-C`, typos in the code, and alarms, and this will lead to confusion. Also, this might catch real errors which should be propagated to the user.

To summarize, only catch specific exceptions as in the following example:

    try:
        return self.__coordinate_ring
    except (AttributeError, OtherExceptions) as msg:           # good
        more_code_to_compute_something()

Note that the syntax in `except` is to list all the exceptions that are caught as a tuple, followed by an error message.

A method or a function accepts input described in the `INPUT` block of the docstring. If the input cannot be handled by the code, then it may raise an exception. The following aims to guide you in choosing from the most relevant exceptions to Sage. Raise

- `TypeError`: if the input belongs to a class of objects that is not supported by the method. For example, a method works only with monic polynomials over a finite field, but a polynomial over rationals was given.

- `ValueError`: if the input has a value not supported by the method. For example, the above method was given a non-monic polynomial.

- `ArithmeticError`: if the method performs an arithmetic operation (sum, product, quotient, and the like) but the input is not appropriate.

- `ZeroDivisionError`: if the method performs division but the input is zero. Note that for non-invertible input values, `ArithmeticError` is more appropriate. As derived from `ArithmeticError`, `ZeroDivisionError` can be caught as `ArithmeticError`.

- `NotImplementedError`: if the input is for a feature not yet implemented by the method. Note that this exception is derived from `RuntimeError`.

If no specific error seems to apply for your situation, `RuntimeError` can be used. In all cases, the string associated with the exception should describe the details of what went wrong.

## Integer return values

Many functions and methods in Sage return integer values. Those should usually be returned as Sage integers of class `Integer` rather than as Python integers of class `int`, as users may want to explore the resulting integers’ number-theoretic properties such as prime factorization. Exceptions should be made when there are good reasons such as performance or compatibility with Python code, for instance in methods such as `__hash__`, `__len__`, and `__int__`.

To return a Python integer `i` as a Sage integer, use:

    from sage.rings.integer import Integer
    return Integer(i)

To return a Sage integer `i` as a Python ineger, use:

    return int(i)

## Importing

We mention two issues with importing: circular imports and importing large third-party modules. See also Dependencies and distribution packages for a discussion of imports from the viewpoint of modularization.

First, you must avoid circular imports. For example, suppose that the file SAGE_ROOT/src/sage/algebras/steenrod_algebra.py started with a line:

    from sage.sage.algebras.steenrod_algebra_bases import *

and that the file SAGE_ROOT/src/sage/algebras/steenrod_algebra_bases.py started with a line:

    from sage.sage.algebras.steenrod_algebra import SteenrodAlgebra

This sets up a loop: loading one of these files requires the other, which then requires the first, etc.

With this set-up, running Sage will produce an error:

    Exception exceptions.ImportError: 'cannot import name SteenrodAlgebra'
    in 'sage.rings.polynomial.polynomial_element.
    Polynomial_generic_dense.__normalize' ignored
    -------------------------------------------------------------------
    ImportError                       Traceback (most recent call last)

    ...
    ImportError: cannot import name SteenrodAlgebra

Instead, you might replace the `import *` line at the top of the file by more specific imports where they are needed in the code. For example, the `basis` method for the class `SteenrodAlgebra` might look like this (omitting the documentation string):

    def basis(self, n):
        from steenrod_algebra_bases import steenrod_algebra_basis
        return steenrod_algebra_basis(n, basis=self._basis_name, p=self.prime)

Second, do not import at the top level of your module a third-party module that will take a long time to initialize (e.g. `matplotlib`). As above, you might instead import specific components of the module when they are needed, rather than at the top level of your file.

It is important to try to make `from sage.all import *` as fast as possible, since this is what dominates the Sage startup time, and controlling the top-level imports helps to do this. One important mechanism in Sage are lazy imports, which don’t actually perform the import but delay it until the object is actually used. See `sage.misc.lazy_import` for more details of lazy imports, and Files and directory structure for an example using lazy imports for a new module.

If your module needs to make some precomputed data available at the top level, you can reduce its load time (and thus startup time, unless your module is imported using `sage.misc.lazy_import`) by using the decorator `sage.misc.cachefunc.cached_function()` instead. For example, replace

    big_data = initialize_big_data()  # bad: runs at module load time

by

    from sage.misc.cachefunc import cached_function

    @cached_function                  # good: runs on first use
    def big_data():
        return initialize_big_data()

## Static typing

Python libraries are increasingly annotated with static typing information; see the Python reference on typing.

For typechecking the Sage library, the project uses pyright; it automatically runs in the GitHub Actions CI and can also be run locally.

As of Sage 10.2, the Sage library only contains a minimal set of such type annotations. Pull requests that add more annotations are generally welcome.

The Sage library makes very extensive use of Cython (see chapter Coding in Cython). Although Cython source code often declares static types for the purpose of compilation to efficient machine code, this typing information is unfortunately not visible to static checkers such as Pyright. It is necessary to create type stub files (“.pyi”) that provide this information. Although various tools for writing and maintaining type stub files are available, creating stub files for Cython files involves manual work. There is hope that better tools become available soon, see for example cython/cython #5744. Contributing to the development and testing of such tools likely will have a greater impact than writing the typestub files manually.

For Cython modules of the Sage library, these type stub files would be placed next to the `.pyx` and `.pxd` files.

When importing from other Python libraries that do not provide sufficient typing information, it is possible to augment the library’s typing information for the purposes of typechecking the Sage library:

- Create typestub files and place them in the directory `SAGE_ROOT/src/typings`. For example, the distribution pplpy provides the top-level package `ppl`, which publishes no typing information. We can create a typestub file `SAGE_ROOT/src/typings/ppl.pyi` or `SAGE_ROOT/src/typings/ppl/__init__.pyi`.

- When these typestub files are working well, it is preferable from the viewpoint of the Sage project that they are “upstreamed”, i.e., contributed to the project that maintains the library. If a new version of the upstream library becomes available that provides the necessary typing information, we can update the package in the Sage distribution and remove the typestub files again from `SAGE_ROOT/src/typings`.

- As a fallback, when neither adding typing annotations to source files nor adding typestub files is welcomed by the upstream project, it is possible to contribute typestubs files instead to the typeshed community project.

## Deprecation

When making a backward-incompatible modification in Sage, the old code should keep working and a message indicating how the code should be updated/written in the future should be displayed somewhere. We call this deprecation. We explain how to do the deprecation, the deprecation policy, below.

Any class, function, method, or attribute defined in a file under SAGE_ROOT/src/sage is subject to the deprecation policy. If its name starts with an underscore, then it is considered internal, and exempt from the deprecation policy.

Note

A deprecated class, function, method, or attribute can only be removed one year after the first stable release in which it appeared.

When a deprecated function, method, or attribute is used, a deprecation warning is issued. The warning message contains the number of the GitHub PR that implemented the deprecation. We use 12345 in the examples below.

Note

For deprecation tools used in the examples, consult the tool’s documentation for more information on its behaviour and optional arguments.

- Rename a keyword: by decorating a function/method with `rename_keyword`, any user calling `my_function(my_old_keyword=5)` will see a warning:

      from sage.misc.decorators import rename_keyword
      @rename_keyword(deprecation=12345, my_old_keyword='my_new_keyword')
      def my_function(my_new_keyword=True):
          return my_new_keyword

- Rename a function/method: call `deprecated_function_alias()` to obtain a copy of a function that raises a deprecation warning:

      from sage.misc.superseded import deprecated_function_alias
      def my_new_function():
          ...

      my_old_function = deprecated_function_alias(12345, my_new_function)

- Moving an object to a different module: if you rename a source file or move some function (or class) to a different file, it should still be possible to import that function from the old module. This can be done using a `lazy_import()` with deprecation. In the old module, you would write:

      from sage.misc.lazy_import import lazy_import
      lazy_import('sage.new.module.name', 'name_of_the_function', deprecation=12345)

  You can also lazily import everything using `*` or a few functions using a tuple:

      from sage.misc.lazy_import import lazy_import
      lazy_import('sage.new.module.name', '*', deprecation=12345)
      lazy_import('sage.other.module', ('func1', 'func2'), deprecation=12345)

- Remove a name from a global namespace: this is when you want to remove a name from a global namespace (say, `sage.all` or some other `all.py` file) but you want to keep the functionality available with an explicit import. This case is similar as the previous one: use a lazy import with deprecation. One detail: in this case, you don’t want the name `lazy_import` to be visible in the global namespace, so we add a leading underscore:

      from sage.misc.lazy_import import lazy_import as _lazy_import
      _lazy_import('sage.some.package', 'some_function', deprecation=12345)

- Any other case: if none of the cases above apply, call `deprecation()` in the function that you want to deprecate. It will display the message of your choice (and interact properly with the doctest framework):

      from sage.misc.superseded import deprecation
      deprecation(12345, "Do not use your computer to compute 1 + 1. Use your brain.")

Note

These decorators only work for Python. There is no implementation of decorators in Cython. Hence, when in need to rename a keyword/function/method/… in a Cython (.pyx) file and/or to deprecate something, forget about decorators and just use `deprecation_cython()` instead. The usage of `deprecation_cython()` is exactly the same as `deprecation()`.

When a class is renamed or removed, it should be deprecated unless it is internal. A class is internal if its name starts with an underscore, or experts (authors and reviewers of a PR making changes to the class) agree that the class is unlikely to be directly imported by user code. Otherwise, or if experts disagree, it is public.

As a class is imported rather than run by user code, there are some technical difficulties in using the above deprecation tools. Instead we follow the procedure below:

- Renaming a class: rename `OldClass` to `NewClass` and add an alias `OldClass = NewClass`:

      class NewClass:
          ...

      OldClass = NewClass   # OldClass is deprecated. See Issue #12345.

- Removing a class: add a comment:

      # OldClass is deprecated. See Issue #12345.

      class OldClass:

In both cases, make it sure to display the change in the “Deprecations” section of the release notes of the next stable release.

## Experimental/unstable code

You can mark your newly created code (classes/functions/methods) as experimental/unstable. In this case, no deprecation warning is needed when changing this code, its functionality or its interface.

This should allow you to put your stuff in Sage early, without worrying about making (design) changes later.

When satisfied with the code (when stable for some time, say, one year), you can delete this warning.

As usual, all code has to be fully doctested and go through our reviewing process.

- Experimental function/method: use the decorator `experimental`. Here is an example:

      from sage.misc.superseded import experimental
      @experimental(12345)
      def experimental_function():
          # do something

- Experimental class: use the decorator `experimental` for its `__init__`. Here is an example:

      from sage.misc.superseded import experimental
      class experimental_class(SageObject):
          @experimental(12345)
          def __init__(self, some, arguments):
              # do something

- Any other case: if none of the cases above apply, call `experimental_warning()` in the code where you want to warn. It will display the message of your choice:

      from sage.misc.superseded import experimental_warning
      experimental_warning(12345, 'This code is not foolproof.')

## Using optional packages

If a function requires an optional package, that function should fail gracefully—perhaps using a `try`-`except` block—when the optional package is not available, and should give a hint about how to install it. For example, typing `sage -optional` gives a list of all optional packages, so it might suggest to the user that they type that. The command `optional_packages()` from within Sage also returns this list.