# Coding in Python for Sage#

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

## Python language standard#

Sage library code needs to be compatible with all versions of Python
that Sage supports. The information regarding the supported versions
can be found in the files `build/pkgs/python3/spkg-configure.m4`

and
`src/setup.cfg.m4`

.

Python 3.9 is the oldest supported version. Hence, all language and library features that are available in Python 3.9 can be used; but features introduced in Python 3.10 cannot 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.Backport packages

importlib_metadata (to be used in place of

`importlib.metadata`

),importlib_resources (to be used in place of

`importlib.resources`

),typing_extensions (to be used in place of

`typing`

).

The Sage library declares these packages as dependencies and ensures that versions that provide features of Python 3.11 are available.

Meta github issue #29756 keeps track of newer Python features and serves as a starting point for discussions on how to make use of them in the Sage library.

## 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.rings.ring import Algebra
class MyFavoriteAlgebra(Algebra):
...
```

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`

.

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

You can use any macros included in

`amsmath`

,`amssymb`

, or`amsfonts`

, or the ones defined in`SAGE_ROOT/doc/commontex/macros.tex`

.

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: R.<x,y> = QQ[] sage: 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: 16.sqrt() 4 sage: 87.factor() 3 * 29

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

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

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:

`SAGE_ROOT/src/bin/sage`

`SAGE_ROOT/src/bin/sage-preparse`

`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: 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
```

Here is another example:

```
sage: R = RealField(10000)
sage: a = R(1) + R(10)^-100
sage: a == RDF(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: hash(Mod(2,7))
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 display a message indicating how it should be updated/written
in the future. We call this a *deprecation*.

Note

Deprecated code can only be removed one year after the first stable release in which it appeared.

Each deprecation warning contains the number of the GitHub PR that defines it. We use 666 in the examples below. For each entry, consult the function’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=666, 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(666, 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=666)

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=666) lazy_import('sage.other.module', ('func1', 'func2'), deprecation=666)

**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=666)

**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(666, "Do not use your computer to compute 1+1. Use your brain.")

### 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(66666) 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(66666) 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(66666, '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.