r"""
Elements
AUTHORS:
- David Harvey (2006-10-16): changed CommutativeAlgebraElement to
derive from CommutativeRingElement instead of AlgebraElement
- David Harvey (2006-10-29): implementation and documentation of new
arithmetic architecture
- William Stein (2006-11): arithmetic architecture -- pushing it
through to completion.
- Gonzalo Tornaria (2007-06): recursive base extend for coercion --
lots of tests
- Robert Bradshaw (2007-2010): arithmetic operators and coercion
- Maarten Derickx (2010-07): added architecture for is_square and sqrt
- Jeroen Demeyer (2016-08): moved all coercion to the base class
:class:`Element`, see :issue:`20767`
The Abstract Element Class Hierarchy
====================================
This is the abstract class hierarchy, i.e., these are all
abstract base classes.
::
SageObject
Element
ModuleElement
RingElement
CommutativeRingElement
IntegralDomainElement
DedekindDomainElement
PrincipalIdealDomainElement
EuclideanDomainElement
FieldElement
CommutativeAlgebraElement
Expression
AlgebraElement
Matrix
InfinityElement
AdditiveGroupElement
Vector
MonoidElement
MultiplicativeGroupElement
ElementWithCachedMethod
How to Define a New Element Class
=================================
Elements typically define a method ``_new_c``, e.g.,
.. code-block:: cython
cdef _new_c(self, defining data):
cdef FreeModuleElement_generic_dense x
x = FreeModuleElement_generic_dense.__new__(FreeModuleElement_generic_dense)
x._parent = self._parent
x._entries = v
that creates a new sibling very quickly from defining data
with assumed properties.
.. _element_arithmetic:
Arithmetic for Elements
-----------------------
Sage has a special system for handling arithmetic operations on Sage
elements (that is instances of :class:`Element`), in particular to
manage uniformly mixed arithmetic operations using the :mod:`coercion
model <sage.structure.coerce>`. We describe here the rules that must
be followed by both arithmetic implementers and callers.
A quick summary for the impatient
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
To implement addition for any :class:`Element` subclass, override the
``def _add_(self, other)`` method instead of the usual Python
``__add__`` :python:`special method <reference/datamodel.html#special-method-names>`.
Within ``_add_(self, other)``, you may assume that ``self`` and
``other`` have the same parent.
If the implementation is generic across all elements in a given
category `C`, then this method can be put in ``C.ElementMethods``.
When writing *Cython* code, ``_add_`` should be a cpdef method:
``cpdef _add_(self, other)``.
When doing arithmetic with two elements having different parents,
the :mod:`coercion model <sage.structure.coerce>` is responsible for
"coercing" them to a common parent and performing arithmetic on the
coerced elements.
Arithmetic in more detail
^^^^^^^^^^^^^^^^^^^^^^^^^
The aims of this system are to provide (1) an efficient calling protocol
from both Python and Cython, (2) uniform coercion semantics across Sage,
(3) ease of use, (4) readability of code.
We will take addition as an example; all other operators are similar.
There are two relevant functions, with differing names
(single vs. double underscores).
- **def Element.__add__(left, right)**
This function is called by Python or Cython when the binary "+"
operator is encountered. It assumes that at least one of its
arguments is an :class:`Element`.
It has a fast pathway to deal with the most common case where both
arguments have the same parent. Otherwise, it uses the coercion
model to work out how to make them have the same parent. The
coercion model then adds the coerced elements (technically, it calls
``operator.add``). Note that the result of coercion is not required
to be a Sage :class:`Element`, it could be a plain Python type.
Note that, although this function is declared as ``def``, it doesn't
have the usual overheads associated with Python functions (either
for the caller or for ``__add__`` itself). This is because Python
has optimised calling protocols for such special functions.
- **def Element._add_(self, other)**
This is the function that you should override to implement addition
in a subclass of :class:`Element`.
The two arguments to this function are guaranteed to have the **same
parent**, but not necessarily the same Python type.
When implementing ``_add_`` in a Cython extension type, use
``cpdef _add_`` instead of ``def _add_``.
In Cython code, if you want to add two elements and you know that
their parents are identical, you are encouraged to call this
function directly, instead of using ``x + y``. This only works if
Cython knows that the left argument is an ``Element``. You can
always cast explicitly: ``(<Element>x)._add_(y)`` to force this.
In plain Python, ``x + y`` is always the fastest way to add two
elements because the special method ``__add__`` is optimized
unlike the normal method ``_add_``.
The difference in the names of the arguments (``left, right``
versus ``self, other``) is intentional: ``self`` is guaranteed to be an
instance of the class in which the method is defined. In Cython, we know
that at least one of ``left`` or ``right`` is an instance of the class
but we do not know a priori which one.
Powering is a special case: first of all, the 3-argument version of
``pow()`` is not supported. Second, the coercion model checks whether
the exponent looks like an integer. If so, the function ``_pow_int``
is called. If the exponent is not an integer, the arguments are coerced
to a common parent and ``_pow_`` is called. So, if your type only
supports powering to an integer exponent, you should implement only
``_pow_int``. If you want to support arbitrary powering, implement both
``_pow_`` and ``_pow_int``.
For addition, multiplication and powering (not for other operators),
there is a fast path for operations with a C ``long``. For example,
implement ``cdef _add_long(self, long n)`` with optimized code for
``self + n``. The addition and multiplication are assumed to be
commutative, so they are also called for ``n + self`` or ``n * self``.
From Cython code, you can also call ``_add_long`` or ``_mul_long``
directly. This is strictly an optimization: there is a default
implementation falling back to the generic arithmetic function.
Examples
^^^^^^^^
We need some :class:`Parent` to work with::
sage: from sage.structure.parent import Parent
sage: class ExampleParent(Parent):
....: def __init__(self, name, **kwds):
....: Parent.__init__(self, **kwds)
....: self.rename(name)
We start with a very basic example of a Python class implementing
``_add_``::
sage: from sage.structure.element import Element
sage: class MyElement(Element):
....: def _add_(self, other):
....: return 42
sage: p = ExampleParent("Some parent")
sage: x = MyElement(p)
sage: x + x
42
When two different parents are involved, this no longer works since
there is no coercion::
sage: q = ExampleParent("Other parent")
sage: y = MyElement(q)
sage: x + y
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +: 'Some parent' and 'Other parent'
If ``_add_`` is not defined, an error message is raised, referring to
the parents::
sage: x = Element(p)
sage: x._add_(x)
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_add_'...
sage: x + x
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +: 'Some parent' and 'Some parent'
sage: y = Element(q)
sage: x + y
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +: 'Some parent' and 'Other parent'
We can also implement arithmetic generically in categories::
sage: class MyCategory(Category):
....: def super_categories(self):
....: return [Sets()]
....: class ElementMethods:
....: def _add_(self, other):
....: return 42
sage: p = ExampleParent("Parent in my category", category=MyCategory())
sage: x = Element(p)
sage: x + x
42
Implementation details
^^^^^^^^^^^^^^^^^^^^^^
Implementing the above features actually takes a bit of magic. Casual
callers and implementers can safely ignore it, but here are the
details for the curious.
To achieve fast arithmetic, it is critical to have a fast path in Cython
to call the ``_add_`` method of a Cython object. So we would like
to declare ``_add_`` as a ``cpdef`` method of class :class:`Element`.
Remember however that the abstract classes coming
from categories come after :class:`Element` in the method resolution
order (or fake method resolution order in case of a Cython
class). Hence any generic implementation of ``_add_`` in such an
abstract class would in principle be shadowed by ``Element._add_``.
This is worked around by defining ``Element._add_`` as a ``cdef``
instead of a ``cpdef`` method. Concrete implementations in subclasses
should be ``cpdef`` or ``def`` methods.
Let us now see what happens upon evaluating ``x + y`` when ``x`` and ``y``
are instances of a class that does not implement ``_add_`` but where
``_add_`` is implemented in the category.
First, ``x.__add__(y)`` is called, where ``__add__`` is implemented
in :class:`Element`.
Assuming that ``x`` and ``y`` have the same parent, a Cython call to
``x._add_(y)`` will be done.
The latter is implemented to trigger a Python level call to ``x._add_(y)``
which will succeed as desired.
In case that Python code calls ``x._add_(y)`` directly,
``Element._add_`` will be invisible, and the method lookup will
continue down the MRO and find the ``_add_`` method in the category.
"""
cimport cython
from cpython cimport *
from sage.ext.stdsage cimport *
cdef add, sub, mul, truediv, floordiv, mod, matmul, pow
from operator import (add, sub, mul, truediv, floordiv, mod, matmul, pow)
cdef dict _coerce_op_symbols = dict(
add='+', sub='-', mul='*', truediv='/', floordiv='//', mod='%', matmul='@', pow='^',
iadd='+', isub='-', imul='*', itruediv='/', ifloordiv='//', imod='%', imatmul='@', ipow='^')
from sage.structure.richcmp cimport rich_to_bool
from sage.structure.coerce cimport py_scalar_to_element, coercion_model
from sage.structure.parent cimport Parent
from sage.cpython.type cimport can_assign_class
from sage.cpython.getattr cimport getattr_from_other_class
from sage.misc.lazy_format import LazyFormat
from sage.arith.long cimport integer_check_long_py
from sage.arith.power cimport generic_power as arith_generic_power
from sage.arith.numerical_approx cimport digits_to_bits
from sage.misc.decorators import sage_wraps
from sage.misc.superseded import deprecation
def make_element(_class, _dict, parent):
"""
This function is only here to support old pickles.
Pickling functionality is moved to Element.{__getstate__,__setstate__}
functions.
"""
from sage.misc.pickle_old import make_element_old
return make_element_old(_class, _dict, parent)
cdef unary_op_exception(op, x):
try:
op = op.__name__
op = _coerce_op_symbols[op]
except (AttributeError, KeyError):
pass
px = parent(x)
return TypeError(f"unsupported operand parent for {op}: '{px}'")
cdef bin_op_exception(op, x, y):
try:
op = op.__name__
op = _coerce_op_symbols[op]
except (AttributeError, KeyError):
pass
px = parent(x)
py = parent(y)
return TypeError(f"unsupported operand parent(s) for {op}: '{px}' and '{py}'")
def is_Element(x):
"""
Return ``True`` if x is of type Element.
EXAMPLES::
sage: from sage.structure.element import is_Element
sage: is_Element(2/3)
doctest:warning...
DeprecationWarning: The function is_Element is deprecated; use 'isinstance(..., Element)' instead.
See https://github.com/sagemath/sage/issues/38077 for details.
True
sage: is_Element(QQ^3) # needs sage.modules
False
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_Element is deprecated; use 'isinstance(..., Element)' instead.")
return isinstance(x, Element)
cdef class Element(SageObject):
"""
Generic element of a structure. All other types of elements
(:class:`RingElement`, :class:`ModuleElement`, etc)
derive from this type.
Subtypes must either call ``__init__()`` to set ``_parent``, or may
set ``_parent`` themselves if that would be more efficient.
.. automethod:: _richcmp_
.. automethod:: __add__
.. automethod:: __sub__
.. automethod:: __neg__
.. automethod:: __mul__
.. automethod:: __truediv__
.. automethod:: __floordiv__
.. automethod:: __mod__
"""
@cython.binding(False)
@cython.always_allow_keywords(False)
def __getmetaclass__(_):
from sage.misc.inherit_comparison import InheritComparisonMetaclass
return InheritComparisonMetaclass
def __init__(self, parent):
r"""
INPUT:
- ``parent`` -- a SageObject
"""
self._parent = parent
def _set_parent(self, parent):
r"""
Set the parent of ``self`` to ``parent``.
INPUT:
- ``parent`` -- a :class:`Parent`
.. WARNING::
Changing the parent of an object is not something you
should normally need. It is mainly meant for constructing a
new element from scratch, when ``__new__`` or ``__init__``
did not set the right parent. Using this method incorrectly
can break things badly.
EXAMPLES::
sage: q = 3/5
sage: parent(q)
Rational Field
sage: q._set_parent(CC) # needs sage.rings.real_mpfr
sage: parent(q) # needs sage.rings.real_mpfr
Complex Field with 53 bits of precision
sage: q._set_parent(float) # needs sage.rings.real_mpfr
Traceback (most recent call last):
...
TypeError: Cannot convert type to sage.structure.parent.Parent
"""
self._parent = <Parent?>parent
def __getattr__(self, name):
"""
Lookup a method or attribute from the category abstract classes.
Let ``P`` be a parent in a category ``C``. Usually the methods
of ``C.element_class`` are made directly available to elements
of ``P`` via standard class inheritance. This is not the case
any more if the elements of ``P`` are instances of an
extension type. See :class:`Category` for details.
The purpose of this method is to emulate this inheritance: for
``e`` and element of ``P``, if an attribute or method
``e.foo`` is not found in the super classes of ``e``, it's
looked up manually in ``C.element_class`` and bound to ``e``.
.. NOTE::
- The attribute or method is actually looked up in
``P._abstract_element_class``. In most cases this is
just an alias for ``C.element_class``, but some parents,
notably homsets, customizes this to let elements also
inherit from other abstract classes. See
:meth:`Parent._abstract_element_class` and
:meth:`Homset._abstract_element_class` for details.
- This mechanism may also enter into action when the
category of `P` is refined on the fly, leaving
previously constructed elements in an outdated element
class.
See :class:`~sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_generic`
for an example.
EXAMPLES:
We test that ``1`` (an instance of the extension type
``Integer``) inherits the methods from the categories of
``ZZ``, that is from ``CommutativeRings().element_class``::
sage: 1.is_idempotent(), 2.is_idempotent()
(True, False)
This method is actually provided by the ``Magmas()`` super
category of ``CommutativeRings()``::
sage: 1.is_idempotent
<bound method Magmas.ElementMethods.is_idempotent of 1>
sage: 1.is_idempotent.__module__
'sage.categories.magmas'
TESTS::
sage: 1.blah_blah
Traceback (most recent call last):
...
AttributeError: 'sage.rings.integer.Integer' object has no attribute 'blah_blah'...
sage: Semigroups().example().an_element().is_idempotent
<bound method LeftZeroSemigroup.Element.is_idempotent of 42>
sage: Semigroups().example().an_element().blah_blah
Traceback (most recent call last):
...
AttributeError: 'LeftZeroSemigroup_with_category.element_class' object has no attribute 'blah_blah'...
"""
return self.getattr_from_category(name)
cdef getattr_from_category(self, name):
cdef Parent P = self._parent
if P is None:
cls = type
else:
cls = P._abstract_element_class
return getattr_from_other_class(self, cls, name)
def __dir__(self):
"""
Emulate ``__dir__`` for elements with dynamically attached methods.
Let cat be the category of the parent of ``self``. This method
emulates ``self`` being an instance of both ``Element`` and
``cat.element_class`` (and the corresponding ``morphism_class`` in the
case of a morphism), in that order, for attribute directory.
EXAMPLES::
sage: dir(1/2)
[..., 'is_idempotent', 'is_integer', 'is_integral', ...]
Caveat: dir on Integer's and some other extension types seem to ignore __dir__::
sage: 1.__dir__()
[..., 'is_idempotent', 'is_integer', 'is_integral', ...]
sage: dir(1) # todo: not implemented
[..., 'is_idempotent', 'is_integer', 'is_integral', ...]
TESTS:
Check that morphism classes are handled correctly (:issue:`29776`)::
sage: R.<x,y> = QQ[]
sage: f = R.hom([x, y+1], R)
sage: 'cartesian_product' in dir(f)
True
sage: 'extend_to_fraction_field' in dir(f)
True
"""
from sage.cpython.getattr import dir_with_other_class
ec = self.parent().category().element_class
try:
mc = self.category_for().morphism_class
except AttributeError:
return dir_with_other_class(self, ec)
else:
return dir_with_other_class(self, ec, mc)
def _repr_(self):
return "Generic element of a structure"
def __getstate__(self):
"""
Return a tuple describing the state of your object.
This should return all information that will be required to unpickle
the object. The functionality for unpickling is implemented in
__setstate__().
TESTS::
sage: R.<x,y> = QQ[]
sage: i = ideal(x^2 - y^2 + 1)
sage: i.__getstate__()
(Monoid of ideals of Multivariate Polynomial Ring in x, y over Rational Field,
{'_Ideal_generic__gens': (x^2 - y^2 + 1,),
'_Ideal_generic__ring': Multivariate Polynomial Ring in x, y over Rational Field,
'_gb_by_ordering': {}})
"""
return (self._parent, self.__dict__)
def __setstate__(self, state):
"""
Initialize the state of the object from data saved in a pickle.
During unpickling ``__init__`` methods of classes are not called, the
saved data is passed to the class via this function instead.
TESTS::
sage: R.<x,y> = QQ[]
sage: i = ideal(x); i
Ideal (x) of Multivariate Polynomial Ring in x, y over Rational Field
sage: S.<x,y,z> = ZZ[]
sage: i.__setstate__((R,{'_Ideal_generic__ring':S,'_Ideal_generic__gens': (S(x^2 - y^2 + 1),)}))
sage: i
Ideal (x^2 - y^2 + 1) of Multivariate Polynomial Ring in x, y, z over Integer Ring
"""
self._set_parent(state[0])
self.__dict__ = state[1]
def __copy__(self):
"""
Return a copy of ``self``.
OUTPUT: a new object which is a copy of ``self``
This implementation ensures that ``self.__dict__`` is properly copied
when it exists (typically for instances of classes deriving from
:class:`Element`).
TESTS::
sage: from sage.structure.element import Element
sage: el = Element(parent = ZZ)
sage: el1 = copy(el)
sage: el1 is el
False
sage: class Demo(Element): pass
sage: el = Demo(parent = ZZ)
sage: el.x = [1,2,3]
sage: el1 = copy(el)
sage: el1 is el
False
sage: el1.__dict__ is el.__dict__
False
"""
cls = self.__class__
cdef Element res = cls.__new__(cls)
res._parent = self._parent
try:
D = self.__dict__
except AttributeError:
return res
for k, v in D.items():
try:
setattr(res, k, v)
except AttributeError:
pass
return res
def _im_gens_(self, codomain, im_gens, base_map=None):
"""
Return the image of ``self`` in codomain under the map that sends
the images of the generators of the parent of ``self`` to the
tuple of elements of im_gens.
"""
raise NotImplementedError
cpdef base_extend(self, R):
cdef Parent V
V = self._parent.base_extend(R)
return V.coerce(self)
def base_ring(self):
"""
Return the base ring of this element's parent (if that makes sense).
TESTS::
sage: QQ.base_ring()
Rational Field
sage: identity_matrix(3).base_ring() # needs sage.modules
Integer Ring
"""
return self._parent.base_ring()
def category(self):
from sage.categories.category_types import Elements
return Elements(self._parent)
def _test_new(self, **options):
"""
Check that ``cls.__new__(cls)`` and
``cls.__new__(cls, parent)`` do not crash Python,
where ``cls = type(self)`` and ``parent = parent(self)``.
It is perfectly legal for ``__new__`` to raise ordinary
exceptions.
EXAMPLES::
sage: from sage.structure.element import Element
sage: p = Parent()
sage: e = Element(p)
sage: e._test_new()
"""
cdef type cls = type(self)
try:
cls.__new__(cls)
except Exception:
pass
try:
cls.__new__(cls, self._parent)
except Exception:
pass
def _test_category(self, **options):
"""
Run generic tests on the method :meth:`.category`.
See also: :class:`TestSuite`.
EXAMPLES::
sage: 3._test_category()
Let us now write a broken :meth:`.category` method::
sage: from sage.categories.examples.sets_cat import PrimeNumbers
sage: class CCls(PrimeNumbers):
....: def an_element(self):
....: return 18
sage: CC = CCls()
sage: CC._test_an_element()
Traceback (most recent call last):
...
AssertionError: self.an_element() is not in self
"""
tester = self._tester(**options)
SageObject._test_category(self, tester=tester)
if can_assign_class(self):
tester.assertTrue(isinstance(self, self.parent().category().element_class))
else:
tester.assertTrue(hasattr(self, "_dummy_attribute"))
def _test_eq(self, **options):
"""
Test that ``self`` is equal to ``self`` and different to ``None``.
See also: :class:`TestSuite`.
TESTS::
sage: from sage.structure.element import Element
sage: O = Element(Parent())
sage: O._test_eq()
Let us now write a broken class method::
sage: class CCls(Element):
....: def __eq__(self, other):
....: return True
sage: CCls(Parent())._test_eq()
Traceback (most recent call last):
...
AssertionError: broken equality: Generic element of a structure == None
Let us now break inequality::
sage: class CCls(Element):
....: def __ne__(self, other):
....: return True
sage: CCls(Parent())._test_eq()
Traceback (most recent call last):
...
AssertionError: broken non-equality: Generic element of a structure != itself
"""
tester = self._tester(**options)
tester.assertTrue(self == self,
LazyFormat("broken equality: %s == itself is False") % self)
tester.assertFalse(self == None,
LazyFormat("broken equality: %s == None") % self)
tester.assertFalse(self != self,
LazyFormat("broken non-equality: %s != itself") % self)
tester.assertTrue(self != None,
LazyFormat("broken non-equality: %s is not != None") % self)
def parent(self, x=None):
"""
Return the parent of this element; or, if the optional argument x is
supplied, the result of coercing x into the parent of this element.
"""
if x is None:
return self._parent
else:
return self._parent(x)
def subs(self, in_dict=None, **kwds):
"""
Substitutes given generators with given values while not touching
other generators.
This is a generic wrapper around ``__call__``. The syntax is
meant to be compatible with the corresponding method for
symbolic expressions.
INPUT:
- ``in_dict`` -- (optional) dictionary of inputs
- ``**kwds`` -- named parameters
OUTPUT: new object if substitution is possible, otherwise ``self``
EXAMPLES::
sage: x, y = PolynomialRing(ZZ,2,'xy').gens()
sage: f = x^2 + y + x^2*y^2 + 5
sage: f((5,y))
25*y^2 + y + 30
sage: f.subs({x:5})
25*y^2 + y + 30
sage: f.subs(x=5)
25*y^2 + y + 30
sage: (1/f).subs(x=5)
1/(25*y^2 + y + 30)
sage: Integer(5).subs(x=4)
5
"""
if not callable(self):
return self
parent = self._parent
try:
ngens = parent.ngens()
except (AttributeError, NotImplementedError, TypeError):
return self
variables = []
for i in range(ngens):
gen = parent.gen(i)
if str(gen) in kwds:
variables.append(kwds[str(gen)])
elif in_dict and gen in in_dict:
variables.append(in_dict[gen])
else:
variables.append(gen)
return self(*variables)
def substitute(self, *args, **kwds):
"""
This calls :meth:`self.subs`.
EXAMPLES::
sage: x, y = PolynomialRing(ZZ, 2, 'xy').gens()
sage: f = x^2 + y + x^2*y^2 + 5
sage: f((5,y))
25*y^2 + y + 30
sage: f.substitute({x: 5})
25*y^2 + y + 30
sage: f.substitute(x=5)
25*y^2 + y + 30
sage: (1/f).substitute(x=5)
1/(25*y^2 + y + 30)
sage: Integer(5).substitute(x=4)
5
"""
return self.subs(*args, **kwds)
def numerical_approx(self, prec=None, digits=None, algorithm=None):
"""
Return a numerical approximation of ``self`` with ``prec`` bits
(or decimal ``digits``) of precision.
No guarantee is made about the accuracy of the result.
INPUT:
- ``prec`` -- precision in bits
- ``digits`` -- precision in decimal digits (only used if
``prec`` is not given)
- ``algorithm`` -- which algorithm to use to compute this
approximation (the accepted algorithms depend on the object)
If neither ``prec`` nor ``digits`` is given, the default
precision is 53 bits (roughly 16 digits).
EXAMPLES::
sage: (2/3).numerical_approx() # needs sage.rings.real_mpfr
0.666666666666667
sage: pi.n(digits=10) # needs sage.symbolic
3.141592654
sage: pi.n(prec=20) # needs sage.symbolic
3.1416
TESTS:
Check that :issue:`14778` is fixed::
sage: (0).n(algorithm='foo') # needs sage.rings.real_mpfr
0.000000000000000
"""
from sage.arith.numerical_approx import numerical_approx_generic
if prec is None:
prec = digits_to_bits(digits)
return numerical_approx_generic(self, prec)
def n(self, prec=None, digits=None, algorithm=None):
"""
Alias for :meth:`numerical_approx`.
EXAMPLES::
sage: (2/3).n() # needs sage.rings.real_mpfr
0.666666666666667
"""
return self.numerical_approx(prec, digits, algorithm)
def _mpmath_(self, prec=53, rounding=None):
"""
Evaluates numerically and returns an mpmath number.
Used as fallback for conversion by mpmath.mpmathify().
.. NOTE::
Currently, the rounding mode is ignored.
EXAMPLES::
sage: # needs mpmath
sage: from sage.libs.mpmath.all import mp, mpmathify
sage: mp.dps = 30
sage: 25._mpmath_(53)
mpf('25.0')
sage: mpmathify(3 + 4*I)
mpc(real='3.0', imag='4.0')
sage: mpmathify(1 + pi) # needs sage.symbolic
mpf('4.14159265358979323846264338327933')
sage: (1 + pi)._mpmath_(10) # needs sage.symbolic
mpf('4.140625')
sage: (1 + pi)._mpmath_(mp.prec) # needs sage.symbolic
mpf('4.14159265358979323846264338327933')
"""
t = self.n(prec)
from sage.rings.real_mpfr import RealNumber
from sage.rings.complex_mpfr import ComplexNumber
if not isinstance(t, (RealNumber, ComplexNumber)):
raise NotImplementedError("mpmath conversion not implemented for %s" % type(self))
return t._mpmath_(prec=prec)
cpdef _act_on_(self, x, bint self_on_left):
"""
Use this method to implement ``self`` acting on ``x``.
Return ``None`` or raise a ``CoercionException`` if no
such action is defined here.
"""
return None
cpdef _acted_upon_(self, x, bint self_on_left):
"""
Use this method to implement ``self`` acted on by x.
Return ``None`` or raise a ``CoercionException`` if no
such action is defined here.
"""
return None
def __xor__(self, right):
raise RuntimeError("Use ** for exponentiation, not '^', which means xor\n"
"in Python, and has the wrong precedence.")
def __pos__(self):
return self
def _coeff_repr(self, no_space=True):
if self._is_atomic():
s = repr(self)
else:
s = "(%s)" % repr(self)
if no_space:
return s.replace(' ', '')
return s
def _latex_coeff_repr(self):
try:
s = self._latex_()
except AttributeError:
s = str(self)
if self._is_atomic():
return s
else:
return "\\left(%s\\right)" % s
def _is_atomic(self):
"""
Return ``True`` if and only if parenthesis are not required when
*printing* out any of `x - s`, `x + s`, `x^s` and `x/s`.
EXAMPLES::
sage: n = 5; n._is_atomic()
True
sage: n = x + 1; n._is_atomic() # needs sage.symbolic
False
"""
if self._parent._repr_option('element_is_atomic'):
return True
s = str(self)
return s.find("+") == -1 and s.find("-") == -1 and s.find(" ") == -1
def __bool__(self):
r"""
Return whether this element is equal to ``self.parent()(0)``.
Note that this is automatically called when converting to
boolean, as in the conditional of an if or while statement.
EXAMPLES::
sage: bool(1) # indirect doctest
True
If ``self.parent()(0)`` raises an exception (because there is no
meaningful zero element,) then this method returns ``True``. Here,
there is no zero morphism of rings that goes to a non-trivial ring::
sage: bool(Hom(ZZ, Zmod(2)).an_element())
True
But there is a zero morphism to the trivial ring::
sage: bool(Hom(ZZ, Zmod(1)).an_element())
False
TESTS:
Verify that :issue:`5185` is fixed::
sage: # needs sage.modules
sage: v = vector({1: 1, 3: -1})
sage: w = vector({1: -1, 3: 1})
sage: v + w
(0, 0, 0, 0)
sage: (v + w).is_zero()
True
sage: bool(v + w)
False
"""
try:
zero = self._parent.zero()
except Exception:
return True
return self != zero
def is_zero(self):
"""
Return ``True`` if ``self`` equals ``self.parent()(0)``.
The default implementation is to fall back to ``not
self.__bool__``.
.. WARNING::
Do not re-implement this method in your subclass but
implement ``__bool__`` instead.
"""
return not self
def _cache_key(self):
"""
Provide a hashable key for an element if it is not hashable.
EXAMPLES::
sage: a = sage.structure.element.Element(ZZ)
sage: a._cache_key()
(Integer Ring, 'Generic element of a structure')
"""
return self.parent(), str(self)
def __richcmp__(self, other, int op):
"""
Compare ``self`` and ``other`` using the coercion framework,
comparing according to the comparison operator ``op``.
Normally, a class will not redefine ``__richcmp__`` but rely on
this ``Element.__richcmp__`` method which uses coercion if
needed to compare elements. After coercion (or if no coercion
is needed), ``_richcmp_`` is called.
If a class wants to implement rich comparison without coercion,
then ``__richcmp__`` should be defined.
See :class:`sage.numerical.linear_functions.LinearConstraint`
for such an example.
For efficiency reasons, a class can do certain "manual"
coercions directly in ``__richcmp__``, using
``coercion_model.richcmp()`` for the remaining cases.
This is done for example in :class:`Integer`.
"""
if have_same_parent(self, other):
return (<Element>self)._richcmp_(other, op)
else:
return coercion_model.richcmp(self, other, op)
cpdef _richcmp_(left, right, int op):
r"""
Basic default implementation of rich comparisons for elements with
equal parents.
It does a comparison by id for ``==`` and ``!=``. Calling this
default method with ``<``, ``<=``, ``>`` or ``>=`` will return
``NotImplemented``.
EXAMPLES::
sage: from sage.structure.parent import Parent
sage: from sage.structure.element import Element
sage: P = Parent()
sage: e1 = Element(P); e2 = Element(P)
sage: e1 == e1 # indirect doctest
True
sage: e1 == e2 # indirect doctest
False
sage: e1 < e2 # indirect doctest
Traceback (most recent call last):
...
TypeError: '<' not supported between instances of 'sage.structure.element.Element' and 'sage.structure.element.Element'
We now create an ``Element`` class where we define ``_richcmp_``
and check that comparison works::
sage: # needs sage.misc.cython
sage: cython(
....: '''
....: from sage.structure.richcmp cimport rich_to_bool
....: from sage.structure.element cimport Element
....: cdef class FloatCmp(Element):
....: cdef float x
....: def __init__(self, float v):
....: self.x = v
....: cpdef _richcmp_(self, other, int op):
....: cdef float x1 = (<FloatCmp>self).x
....: cdef float x2 = (<FloatCmp>other).x
....: return rich_to_bool(op, (x1 > x2) - (x1 < x2))
....: ''')
sage: a = FloatCmp(1)
sage: b = FloatCmp(2)
sage: a <= b, b <= a
(True, False)
"""
if left is right:
return rich_to_bool(op, 0)
if op == Py_EQ:
return False
if op == Py_NE:
return True
return NotImplemented
def __add__(left, right):
"""
Top-level addition operator for :class:`Element` invoking
the coercion model.
See :ref:`element_arithmetic`.
EXAMPLES::
sage: from sage.structure.element import Element
sage: class MyElement(Element):
....: def _add_(self, other):
....: return 42
sage: e = MyElement(Parent())
sage: e + e
42
TESTS::
sage: e = Element(Parent())
sage: e + e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +: '<sage.structure.parent.Parent object at ...>' and '<sage.structure.parent.Parent object at ...>'
sage: 1 + e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +: 'Integer Ring' and '<sage.structure.parent.Parent object at ...>'
sage: e + 1
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +: '<sage.structure.parent.Parent object at ...>' and 'Integer Ring'
sage: int(1) + e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for +: 'int' and 'sage.structure.element.Element'
sage: e + int(1)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for +: 'sage.structure.element.Element' and 'int'
sage: None + e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for +: 'NoneType' and 'sage.structure.element.Element'
sage: e + None
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for +: 'sage.structure.element.Element' and 'NoneType'
"""
cdef int cl = classify_elements(left, right)
if HAVE_SAME_PARENT(cl):
return (<Element>left)._add_(right)
if BOTH_ARE_ELEMENT(cl):
return coercion_model.bin_op(left, right, add)
cdef long value
cdef int err = -1
try:
integer_check_long_py(right, &value, &err)
if not err:
return (<Element>left)._add_long(value)
integer_check_long_py(left, &value, &err)
if not err:
return (<Element>right)._add_long(value)
return coercion_model.bin_op(left, right, add)
except TypeError:
return NotImplemented
cdef _add_(self, other):
"""
Virtual addition method for elements with identical parents.
This default Cython implementation of ``_add_`` calls the
Python method ``self._add_`` if it exists. This method may be
defined in the ``ElementMethods`` of the category of the parent.
If the method is not found, a :exc:`TypeError` is raised
indicating that the operation is not supported.
See :ref:`element_arithmetic`.
EXAMPLES:
This method is not visible from Python::
sage: from sage.structure.element import Element
sage: e = Element(Parent())
sage: e._add_(e)
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_add_'...
"""
try:
python_op = (<object>self)._add_
except AttributeError:
raise bin_op_exception('+', self, other)
else:
return python_op(other)
cdef _add_long(self, long n):
"""
Generic path for adding a C long, assumed to commute.
EXAMPLES::
sage: # needs sage.misc.cython
sage: cython( # long time
....: '''
....: from sage.structure.element cimport Element
....: cdef class MyElement(Element):
....: cdef _add_long(self, long n):
....: return n
....: ''')
sage: e = MyElement(Parent()) # long time
sage: i = int(42)
sage: i + e, e + i # long time
(42, 42)
"""
return coercion_model.bin_op(self, n, add)
def __sub__(left, right):
"""
Top-level subtraction operator for :class:`Element` invoking
the coercion model.
See :ref:`element_arithmetic`.
EXAMPLES::
sage: from sage.structure.element import Element
sage: class MyElement(Element):
....: def _sub_(self, other):
....: return 42
sage: e = MyElement(Parent())
sage: e - e
42
TESTS::
sage: e = Element(Parent())
sage: e - e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for -: '<sage.structure.parent.Parent object at ...>' and '<sage.structure.parent.Parent object at ...>'
sage: 1 - e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for -: 'Integer Ring' and '<sage.structure.parent.Parent object at ...>'
sage: e - 1
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for -: '<sage.structure.parent.Parent object at ...>' and 'Integer Ring'
sage: int(1) - e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for -: 'int' and 'sage.structure.element.Element'
sage: e - int(1)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for -: 'sage.structure.element.Element' and 'int'
sage: None - e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for -: 'NoneType' and 'sage.structure.element.Element'
sage: e - None
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for -: 'sage.structure.element.Element' and 'NoneType'
"""
cdef int cl = classify_elements(left, right)
if HAVE_SAME_PARENT(cl):
return (<Element>left)._sub_(right)
if BOTH_ARE_ELEMENT(cl):
return coercion_model.bin_op(left, right, sub)
try:
return coercion_model.bin_op(left, right, sub)
except TypeError:
return NotImplemented
cdef _sub_(self, other):
"""
Virtual subtraction method for elements with identical parents.
This default Cython implementation of ``_sub_`` calls the
Python method ``self._sub_`` if it exists. This method may be
defined in the ``ElementMethods`` of the category of the parent.
If the method is not found, a :exc:`TypeError` is raised
indicating that the operation is not supported.
See :ref:`element_arithmetic`.
EXAMPLES:
This method is not visible from Python::
sage: from sage.structure.element import Element
sage: e = Element(Parent())
sage: e._sub_(e)
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_sub_'...
"""
try:
python_op = (<object>self)._sub_
except AttributeError:
raise bin_op_exception('-', self, other)
else:
return python_op(other)
def __neg__(self):
"""
Top-level negation operator for :class:`Element`.
EXAMPLES::
sage: from sage.structure.element import Element
sage: class MyElement(Element):
....: def _neg_(self):
....: return 42
sage: e = MyElement(Parent())
sage: -e
42
TESTS::
sage: e = Element(Parent())
sage: -e
Traceback (most recent call last):
...
TypeError: unsupported operand parent for unary -: '<sage.structure.parent.Parent object at ...>'
"""
return self._neg_()
cdef _neg_(self):
"""
Virtual unary negation method for elements.
This default Cython implementation of ``_neg_`` calls the
Python method ``self._neg_`` if it exists. This method may be
defined in the ``ElementMethods`` of the category of the parent.
If the method is not found, a :exc:`TypeError` is raised
indicating that the operation is not supported.
See :ref:`element_arithmetic`.
EXAMPLES:
This method is not visible from Python::
sage: from sage.structure.element import Element
sage: e = Element(Parent())
sage: e._neg_()
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_neg_'...
"""
try:
python_op = (<object>self)._neg_
except AttributeError:
raise unary_op_exception('unary -', self)
else:
return python_op()
def __mul__(left, right):
"""
Top-level multiplication operator for :class:`Element` invoking
the coercion model.
See :ref:`element_arithmetic`.
EXAMPLES::
sage: from sage.structure.element import Element
sage: class MyElement(Element):
....: def _mul_(self, other):
....: return 42
sage: e = MyElement(Parent())
sage: e * e
42
TESTS::
sage: e = Element(Parent())
sage: e * e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: '<sage.structure.parent.Parent object at ...>' and '<sage.structure.parent.Parent object at ...>'
sage: 1 * e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: 'Integer Ring' and '<sage.structure.parent.Parent object at ...>'
sage: e * 1
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: '<sage.structure.parent.Parent object at ...>' and 'Integer Ring'
sage: int(1) * e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for *: 'int' and 'sage.structure.element.Element'
sage: e * int(1)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for *: 'sage.structure.element.Element' and 'int'
sage: None * e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for *: 'NoneType' and 'sage.structure.element.Element'
sage: e * None
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for *: 'sage.structure.element.Element' and 'NoneType'
::
sage: # needs sage.combinat sage.modules
sage: A = AlgebrasWithBasis(QQ).example(); A
An example of an algebra with basis: the free algebra
on the generators ('a', 'b', 'c') over Rational Field
sage: x = A.an_element()
sage: x
B[word: ] + 2*B[word: a] + 3*B[word: b] + B[word: bab]
sage: x.__mul__(x)
B[word: ] + 4*B[word: a] + 4*B[word: aa] + 6*B[word: ab]
+ 2*B[word: abab] + 6*B[word: b] + 6*B[word: ba]
+ 2*B[word: bab] + 2*B[word: baba] + 3*B[word: babb]
+ B[word: babbab] + 9*B[word: bb] + 3*B[word: bbab]
"""
cdef int cl = classify_elements(left, right)
if HAVE_SAME_PARENT(cl):
return (<Element>left)._mul_(right)
if BOTH_ARE_ELEMENT(cl):
return coercion_model.bin_op(left, right, mul)
cdef long value
cdef int err = -1
try:
integer_check_long_py(right, &value, &err)
if not err:
return (<Element>left)._mul_long(value)
integer_check_long_py(left, &value, &err)
if not err:
return (<Element>right)._mul_long(value)
return coercion_model.bin_op(left, right, mul)
except TypeError:
return NotImplemented
cdef _mul_(self, other):
"""
Virtual multiplication method for elements with identical parents.
This default Cython implementation of ``_mul_`` calls the
Python method ``self._mul_`` if it exists. This method may be
defined in the ``ElementMethods`` of the category of the parent.
If the method is not found, a :exc:`TypeError` is raised
indicating that the operation is not supported.
See :ref:`element_arithmetic`.
EXAMPLES:
This method is not visible from Python::
sage: from sage.structure.element import Element
sage: e = Element(Parent())
sage: e._mul_(e)
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_mul_'...
"""
try:
python_op = (<object>self)._mul_
except AttributeError:
raise bin_op_exception('*', self, other)
else:
return python_op(other)
cdef _mul_long(self, long n):
"""
Generic path for multiplying by a C long, assumed to commute.
EXAMPLES::
sage: # needs sage.misc.cython
sage: cython( # long time
....: '''
....: from sage.structure.element cimport Element
....: cdef class MyElement(Element):
....: cdef _mul_long(self, long n):
....: return n
....: ''')
sage: e = MyElement(Parent()) # long time
sage: i = int(42)
sage: i * e, e * i # long time
(42, 42)
"""
return coercion_model.bin_op(self, n, mul)
def __matmul__(left, right):
"""
Top-level matrix multiplication operator for :class:`Element`
invoking the coercion model.
See :ref:`element_arithmetic`.
EXAMPLES::
sage: from sage.structure.element import Element
sage: class MyElement(Element):
....: def _matmul_(self, other):
....: return 42
sage: e = MyElement(Parent())
sage: from operator import matmul
sage: matmul(e, e)
42
TESTS::
sage: e = Element(Parent())
sage: matmul(e, e)
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for @: '<sage.structure.parent.Parent object at ...>' and '<sage.structure.parent.Parent object at ...>'
sage: matmul(1, e)
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for @: 'Integer Ring' and '<sage.structure.parent.Parent object at ...>'
sage: matmul(e, 1)
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for @: '<sage.structure.parent.Parent object at ...>' and 'Integer Ring'
sage: matmul(int(1), e)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for @: 'int' and 'sage.structure.element.Element'
sage: matmul(e, int(1))
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for @: 'sage.structure.element.Element' and 'int'
sage: matmul(None, e)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for @: 'NoneType' and 'sage.structure.element.Element'
sage: matmul(e, None)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for @: 'sage.structure.element.Element' and 'NoneType'
"""
cdef int cl = classify_elements(left, right)
if HAVE_SAME_PARENT(cl):
return (<Element>left)._matmul_(right)
if BOTH_ARE_ELEMENT(cl):
return coercion_model.bin_op(left, right, matmul)
try:
return coercion_model.bin_op(left, right, matmul)
except TypeError:
return NotImplemented
cdef _matmul_(self, other):
"""
Virtual matrix multiplication method for elements with
identical parents.
This default Cython implementation of ``_matmul_`` calls the
Python method ``self._matmul_`` if it exists. This method may
be defined in the ``ElementMethods`` of the category of the
parent. If the method is not found, a :exc:`TypeError` is raised
indicating that the operation is not supported.
See :ref:`element_arithmetic`.
EXAMPLES:
This method is not visible from Python::
sage: from sage.structure.element import Element
sage: e = Element(Parent())
sage: e._matmul_(e)
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_matmul_'...
"""
try:
python_op = (<object>self)._matmul_
except AttributeError:
raise bin_op_exception('@', self, other)
else:
return python_op(other)
def __truediv__(left, right):
"""
Top-level true division operator for :class:`Element` invoking
the coercion model.
See :ref:`element_arithmetic`.
EXAMPLES::
sage: operator.truediv(2, 3)
2/3
sage: operator.truediv(pi, 3) # needs sage.symbolic
1/3*pi
sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1) # needs sage.rings.number_field
sage: operator.truediv(2, K.ideal(i + 1)) # needs sage.rings.number_field
Fractional ideal (-i + 1)
::
sage: from sage.structure.element import Element
sage: class MyElement(Element):
....: def _div_(self, other):
....: return 42
sage: e = MyElement(Parent())
sage: operator.truediv(e, e)
42
TESTS::
sage: e = Element(Parent())
sage: operator.truediv(e, e)
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for /: '<sage.structure.parent.Parent object at ...>' and '<sage.structure.parent.Parent object at ...>'
sage: operator.truediv(1, e)
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for /: 'Integer Ring' and '<sage.structure.parent.Parent object at ...>'
sage: operator.truediv(e, 1)
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for /: '<sage.structure.parent.Parent object at ...>' and 'Integer Ring'
sage: operator.truediv(int(1), e)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for /: 'int' and 'sage.structure.element.Element'
sage: operator.truediv(e, int(1))
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for /: 'sage.structure.element.Element' and 'int'
sage: operator.truediv(None, e)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for /: 'NoneType' and 'sage.structure.element.Element'
sage: operator.truediv(e, None)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for /: 'sage.structure.element.Element' and 'NoneType'
"""
cdef int cl = classify_elements(left, right)
if HAVE_SAME_PARENT(cl):
return (<Element>left)._div_(right)
if BOTH_ARE_ELEMENT(cl):
return coercion_model.bin_op(left, right, truediv)
try:
return coercion_model.bin_op(left, right, truediv)
except TypeError:
return NotImplemented
cdef _div_(self, other):
"""
Virtual division method for elements with identical parents.
This is called for Python 2 division as well as true division.
This default Cython implementation of ``_div_`` calls the
Python method ``self._div_`` if it exists. This method may be
defined in the ``ElementMethods`` of the category of the parent.
If the method is not found, a :exc:`TypeError` is raised
indicating that the operation is not supported.
See :ref:`element_arithmetic`.
EXAMPLES:
This method is not visible from Python::
sage: from sage.structure.element import Element
sage: e = Element(Parent())
sage: e._div_(e)
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_div_'...
"""
try:
python_op = (<object>self)._div_
except AttributeError:
raise bin_op_exception('/', self, other)
else:
return python_op(other)
def __floordiv__(left, right):
"""
Top-level floor division operator for :class:`Element` invoking
the coercion model.
See :ref:`element_arithmetic`.
EXAMPLES::
sage: 7 // 3
2
sage: 7 // int(3)
2
sage: int(7) // 3
2
::
sage: from sage.structure.element import Element
sage: class MyElement(Element):
....: def _floordiv_(self, other):
....: return 42
sage: e = MyElement(Parent())
sage: e // e
42
TESTS::
sage: e = Element(Parent())
sage: e // e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for //: '<sage.structure.parent.Parent object at ...>' and '<sage.structure.parent.Parent object at ...>'
sage: 1 // e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for //: 'Integer Ring' and '<sage.structure.parent.Parent object at ...>'
sage: e // 1
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for //: '<sage.structure.parent.Parent object at ...>' and 'Integer Ring'
sage: int(1) // e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for //: 'int' and 'sage.structure.element.Element'
sage: e // int(1)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for //: 'sage.structure.element.Element' and 'int'
sage: None // e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for //: 'NoneType' and 'sage.structure.element.Element'
sage: e // None
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for //: 'sage.structure.element.Element' and 'NoneType'
"""
cdef int cl = classify_elements(left, right)
if HAVE_SAME_PARENT(cl):
return (<Element>left)._floordiv_(right)
if BOTH_ARE_ELEMENT(cl):
return coercion_model.bin_op(left, right, floordiv)
try:
return coercion_model.bin_op(left, right, floordiv)
except TypeError:
return NotImplemented
cdef _floordiv_(self, other):
"""
Virtual floor division method for elements with identical parents.
This default Cython implementation of ``_floordiv_`` calls the
Python method ``self._floordiv_`` if it exists. This method may be
defined in the ``ElementMethods`` of the category of the parent.
If the method is not found, a :exc:`TypeError` is raised
indicating that the operation is not supported.
See :ref:`element_arithmetic`.
EXAMPLES:
This method is not visible from Python::
sage: from sage.structure.element import Element
sage: e = Element(Parent())
sage: e._floordiv_(e)
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_floordiv_'...
"""
try:
python_op = (<object>self)._floordiv_
except AttributeError:
raise bin_op_exception('//', self, other)
else:
return python_op(other)
def __mod__(left, right):
"""
Top-level modulo operator for :class:`Element` invoking
the coercion model.
See :ref:`element_arithmetic`.
EXAMPLES::
sage: 7 % 3
1
sage: 7 % int(3)
1
sage: int(7) % 3
1
::
sage: from sage.structure.element import Element
sage: class MyElement(Element):
....: def _mod_(self, other):
....: return 42
sage: e = MyElement(Parent())
sage: e % e
42
TESTS::
sage: e = Element(Parent())
sage: e % e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for %: '<sage.structure.parent.Parent object at ...>' and '<sage.structure.parent.Parent object at ...>'
sage: 1 % e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for %: 'Integer Ring' and '<sage.structure.parent.Parent object at ...>'
sage: e % 1
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for %: '<sage.structure.parent.Parent object at ...>' and 'Integer Ring'
sage: int(1) % e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for %: 'int' and 'sage.structure.element.Element'
sage: e % int(1)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for %: 'sage.structure.element.Element' and 'int'
sage: None % e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for %: 'NoneType' and 'sage.structure.element.Element'
sage: e % None
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for %: 'sage.structure.element.Element' and 'NoneType'
"""
cdef int cl = classify_elements(left, right)
if HAVE_SAME_PARENT(cl):
return (<Element>left)._mod_(right)
if BOTH_ARE_ELEMENT(cl):
return coercion_model.bin_op(left, right, mod)
try:
return coercion_model.bin_op(left, right, mod)
except TypeError:
return NotImplemented
cdef _mod_(self, other):
"""
Virtual modulo method for elements with identical parents.
This default Cython implementation of ``_mod_`` calls the
Python method ``self._mod_`` if it exists. This method may be
defined in the ``ElementMethods`` of the category of the parent.
If the method is not found, a :exc:`TypeError` is raised
indicating that the operation is not supported.
See :ref:`element_arithmetic`.
EXAMPLES:
This method is not visible from Python::
sage: from sage.structure.element import Element
sage: e = Element(Parent())
sage: e._mod_(e)
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_mod_'...
"""
try:
python_op = (<object>self)._mod_
except AttributeError:
raise bin_op_exception('%', self, other)
else:
return python_op(other)
def __pow__(left, right, modulus):
"""
Top-level power operator for :class:`Element` invoking
the coercion model.
See :ref:`element_arithmetic`.
EXAMPLES::
sage: from sage.structure.element import Element
sage: class MyElement(Element):
....: def _add_(self, other):
....: return 42
sage: e = MyElement(Parent())
sage: e + e
42
sage: a = Integers(389)['x']['y'](37)
sage: p = sage.structure.element.RingElement.__pow__
sage: p(a, 2)
202
sage: p(a, 2, 1)
Traceback (most recent call last):
...
TypeError: the 3-argument version of pow() is not supported
::
sage: # needs sage.symbolic
sage: (2/3)^I
(2/3)^I
sage: (2/3)^sqrt(2)
(2/3)^sqrt(2)
sage: var('x,y,z,n')
(x, y, z, n)
sage: (2/3)^(x^n + y^n + z^n)
(2/3)^(x^n + y^n + z^n)
sage: (-7/11)^(tan(x)+exp(x))
(-7/11)^(e^x + tan(x))
sage: float(1.2)**(1/2)
1.0954451150103321
sage: complex(1,2)**(1/2) # needs sage.rings.complex_double
(1.272019649514069+0.786151377757423...j)
TESTS::
sage: e = Element(Parent())
sage: e ^ e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for ^: '<sage.structure.parent.Parent object at ...>' and '<sage.structure.parent.Parent object at ...>'
sage: 1 ^ e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for ^: 'Integer Ring' and '<sage.structure.parent.Parent object at ...>'
sage: e ^ 1
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for ^: '<sage.structure.parent.Parent object at ...>' and 'Integer Ring'
sage: int(1) ^ e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for ** or pow(): 'int' and 'sage.structure.element.Element'
sage: e ^ int(1)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for ** or pow(): 'sage.structure.element.Element' and 'int'
sage: None ^ e
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for ** or pow(): 'NoneType' and 'sage.structure.element.Element'
sage: e ^ None
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for ** or pow(): 'sage.structure.element.Element' and 'NoneType'
"""
if modulus is not None:
raise TypeError("the 3-argument version of pow() is not supported")
cdef int cl = classify_elements(left, right)
if HAVE_SAME_PARENT(cl):
return (<Element>left)._pow_(right)
if BOTH_ARE_ELEMENT(cl):
return coercion_model.bin_op(left, right, pow)
cdef long value
cdef int err = -1
try:
integer_check_long_py(right, &value, &err)
if not err:
return (<Element>left)._pow_long(value)
return coercion_model.bin_op(left, right, pow)
except TypeError:
return NotImplemented
cdef _pow_(self, other):
"""
Virtual powering method for elements with identical parents.
This default Cython implementation of ``_pow_`` calls the
Python method ``self._pow_`` if it exists. This method may be
defined in the ``ElementMethods`` of the category of the parent.
If the method is not found, a :exc:`TypeError` is raised
indicating that the operation is not supported.
See :ref:`element_arithmetic`.
EXAMPLES:
This method is not visible from Python::
sage: from sage.structure.element import Element
sage: e = Element(Parent())
sage: e._pow_(e)
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_pow_'...
"""
try:
python_op = (<object>self)._pow_
except AttributeError:
raise bin_op_exception('^', self, other)
else:
return python_op(other)
cdef _pow_int(self, other):
"""
Virtual powering method for powering to an integer exponent.
This default Cython implementation of ``_pow_int`` calls the
Python method ``self._pow_int`` if it exists. This method may be
defined in the ``ElementMethods`` of the category of the parent.
If the method is not found, a :exc:`TypeError` is raised
indicating that the operation is not supported.
See :ref:`element_arithmetic`.
EXAMPLES:
This method is not visible from Python::
sage: from sage.structure.element import Element
sage: e = Element(Parent())
sage: e._pow_int(e)
Traceback (most recent call last):
...
AttributeError: 'sage.structure.element.Element' object has no attribute '_pow_int'...
"""
try:
python_op = (<object>self)._pow_int
except AttributeError:
raise bin_op_exception('^', self, other)
else:
return python_op(other)
cdef _pow_long(self, long n):
"""
Generic path for powering with a C long.
"""
return self._pow_int(n)
def is_ModuleElement(x):
"""
Return ``True`` if x is of type ModuleElement.
This is even faster than using isinstance inline.
EXAMPLES::
sage: from sage.structure.element import is_ModuleElement
sage: is_ModuleElement(2/3)
doctest:warning...
DeprecationWarning: The function is_ModuleElement is deprecated; use 'isinstance(..., ModuleElement)' instead.
See https://github.com/sagemath/sage/issues/38077 for details.
True
sage: is_ModuleElement((QQ^3).0) # needs sage.modules
True
sage: is_ModuleElement('a')
False
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_ModuleElement is deprecated; use 'isinstance(..., ModuleElement)' instead.")
return isinstance(x, ModuleElement)
cdef class ElementWithCachedMethod(Element):
r"""
An element class that fully supports cached methods.
NOTE:
The :class:`~sage.misc.cachefunc.cached_method` decorator provides
a convenient way to automatically cache the result of a computation.
Since :issue:`11115`, the cached method decorator applied to a
method without optional arguments is faster than a hand-written cache
in Python, and a cached method without any arguments (except ``self``)
is actually faster than a Python method that does nothing more but
to return ``1``. A cached method can also be inherited from the parent
or element class of a category.
However, this holds true only if attribute assignment is supported.
If you write an extension class in Cython that does not accept attribute
assignment then a cached method inherited from the category will be
slower (for :class:`~sage.structure.parent.Parent`) or the cache would
even break (for :class:`Element`).
This class should be used if you write an element class, cannot provide
it with attribute assignment, but want that it inherits a cached method
from the category. Under these conditions, your class should inherit
from this class rather than :class:`Element`. Then, the cache will work,
but certainly slower than with attribute assignment. Lazy attributes
work as well.
EXAMPLES:
We define three element extension classes. The first inherits from
:class:`Element`, the second from this class, and the third simply
is a Python class. We also define a parent class and, in Python, a
category whose element and parent classes define cached methods.
::
sage: # needs sage.misc.cython
sage: cython_code = ["from sage.structure.element cimport Element, ElementWithCachedMethod",
....: "from sage.structure.richcmp cimport richcmp",
....: "cdef class MyBrokenElement(Element):",
....: " cdef public object x",
....: " def __init__(self, P, x):",
....: " self.x = x",
....: " Element.__init__(self, P)",
....: " def __neg__(self):",
....: " return MyBrokenElement(self.parent(), -self.x)",
....: " def _repr_(self):",
....: " return '<%s>' % self.x",
....: " def __hash__(self):",
....: " return hash(self.x)",
....: " cpdef _richcmp_(left, right, int op):",
....: " return richcmp(left.x, right.x, op)",
....: " def raw_test(self):",
....: " return -self",
....: "cdef class MyElement(ElementWithCachedMethod):",
....: " cdef public object x",
....: " def __init__(self, P, x):",
....: " self.x = x",
....: " Element.__init__(self, P)",
....: " def __neg__(self):",
....: " return MyElement(self.parent(), -self.x)",
....: " def _repr_(self):",
....: " return '<%s>' % self.x",
....: " def __hash__(self):",
....: " return hash(self.x)",
....: " cpdef _richcmp_(left, right, int op):",
....: " return richcmp(left.x, right.x, op)",
....: " def raw_test(self):",
....: " return -self",
....: "class MyPythonElement(MyBrokenElement): pass",
....: "from sage.structure.parent cimport Parent",
....: "cdef class MyParent(Parent):",
....: " Element = MyElement"]
sage: cython('\n'.join(cython_code))
sage: cython_code = ["from sage.misc.cachefunc import cached_method",
....: "from sage.misc.cachefunc import cached_in_parent_method",
....: "from sage.categories.category import Category",
....: "from sage.categories.objects import Objects",
....: "class MyCategory(Category):",
....: " @cached_method",
....: " def super_categories(self):",
....: " return [Objects()]",
....: " class ElementMethods:",
....: " @cached_method",
....: " def element_cache_test(self):",
....: " return -self",
....: " @cached_in_parent_method",
....: " def element_via_parent_test(self):",
....: " return -self",
....: " class ParentMethods:",
....: " @cached_method",
....: " def one(self):",
....: " return self.element_class(self,1)",
....: " @cached_method",
....: " def invert(self, x):",
....: " return -x"]
sage: cython('\n'.join(cython_code))
sage: C = MyCategory()
sage: P = MyParent(category=C)
sage: ebroken = MyBrokenElement(P, 5)
sage: e = MyElement(P, 5)
The cached methods inherited by ``MyElement`` works::
sage: # needs sage.misc.cython
sage: e.element_cache_test()
<-5>
sage: e.element_cache_test() is e.element_cache_test()
True
sage: e.element_via_parent_test()
<-5>
sage: e.element_via_parent_test() is e.element_via_parent_test()
True
The other element class can only inherit a
``cached_in_parent_method``, since the cache is stored in the
parent. In fact, equal elements share the cache, even if they are
of different types::
sage: e == ebroken # needs sage.misc.cython
True
sage: type(e) == type(ebroken) # needs sage.misc.cython
False
sage: ebroken.element_via_parent_test() is e.element_via_parent_test() # needs sage.misc.cython
True
However, the cache of the other inherited method breaks, although the method
as such works::
sage: ebroken.element_cache_test() # needs sage.misc.cython
<-5>
sage: ebroken.element_cache_test() is ebroken.element_cache_test() # needs sage.misc.cython
False
Since ``e`` and ``ebroken`` share the cache, when we empty it for one element
it is empty for the other as well::
sage: b = ebroken.element_via_parent_test() # needs sage.misc.cython
sage: e.element_via_parent_test.clear_cache() # needs sage.misc.cython
sage: b is ebroken.element_via_parent_test() # needs sage.misc.cython
False
Note that the cache only breaks for elements that do no allow attribute assignment.
A Python version of ``MyBrokenElement`` therefore allows for cached methods::
sage: epython = MyPythonElement(P, 5) # needs sage.misc.cython
sage: epython.element_cache_test() # needs sage.misc.cython
<-5>
sage: epython.element_cache_test() is epython.element_cache_test() # needs sage.misc.cython
True
"""
cdef getattr_from_category(self, name):
"""
This getattr method ensures that cached methods and lazy attributes
can be inherited from the element class of a category.
.. NOTE::
The use of cached methods is demonstrated in the main doc
string of this class. Here, we demonstrate lazy
attributes.
EXAMPLES::
sage: # needs sage.misc.cython
sage: cython(
....: '''
....: from sage.structure.element cimport ElementWithCachedMethod
....: cdef class MyElement(ElementWithCachedMethod):
....: cdef public object x
....: def __init__(self, P, x):
....: self.x = x
....: ElementWithCachedMethod.__init__(self,P)
....: def _repr_(self):
....: return '<%s>'%self.x
....: from sage.structure.parent cimport Parent
....: cdef class MyParent(Parent):
....: Element = MyElement
....: from sage.misc.cachefunc import cached_method
....: from sage.misc.lazy_attribute import lazy_attribute
....: from sage.categories.category import Category
....: from sage.categories.objects import Objects
....: class MyCategory(Category):
....: @cached_method
....: def super_categories(self):
....: return [Objects()]
....: class ElementMethods:
....: @lazy_attribute
....: def my_lazy_attr(self):
....: return 'lazy attribute of <%s>'%self.x
....: ''')
sage: C = MyCategory()
sage: P = MyParent(category=C)
sage: e = MyElement(P, 5)
sage: e.my_lazy_attr
'lazy attribute of <5>'
sage: e.my_lazy_attr is e.my_lazy_attr
True
"""
try:
return self._cached_methods[name]
except KeyError:
attr = getattr_from_other_class(self,
self._parent.category().element_class,
name)
self._cached_methods[name] = attr
return attr
except TypeError:
attr = getattr_from_other_class(self,
self._parent.category().element_class,
name)
self._cached_methods = {name: attr}
return attr
cdef class ModuleElement(Element):
"""
Generic element of a module.
"""
cpdef _add_(self, other):
"""
Abstract addition method.
TESTS::
sage: from sage.structure.element import ModuleElement
sage: e = ModuleElement(Parent())
sage: e + e
Traceback (most recent call last):
...
NotImplementedError: addition not implemented for <sage.structure.parent.Parent object at ...>
"""
raise NotImplementedError(f"addition not implemented for {self._parent}")
cdef _add_long(self, long n):
"""
Generic path for adding a C long, assumed to commute.
"""
if n == 0:
return self
return coercion_model.bin_op(self, n, add)
cpdef _sub_(self, other):
"""
Default implementation of subtraction using addition and
negation.
"""
return self + (-other)
cpdef _neg_(self):
"""
Default implementation of negation using multiplication
with -1.
"""
return self._mul_long(-1)
cdef _mul_long(self, long n):
"""
Generic path for multiplying by a C long, assumed to commute.
"""
if n == 1:
return self
return coercion_model.bin_op(self, n, mul)
cpdef _rmul_(self, Element left):
"""
Reversed scalar multiplication for module elements with the
module element on the right and the scalar on the left.
By default, we assume commutativity and reverse the arguments.
"""
return self._lmul_(left)
cpdef _lmul_(self, Element right):
"""
Scalar multiplication for module elements with the module
element on the left and the scalar on the right.
Returning ``None`` indicates that this action is not implemented here.
"""
return None
def order(self):
"""
Return the additive order of ``self``.
"""
return self.additive_order()
def additive_order(self):
"""
Return the additive order of ``self``.
"""
raise NotImplementedError
cdef class ModuleElementWithMutability(ModuleElement):
"""
Generic element of a module with mutability.
"""
def __init__(self, parent, is_immutable=False):
"""
EXAMPLES::
sage: v = sage.modules.free_module_element.FreeModuleElement(QQ^3) # needs sage.modules
sage: type(v) # needs sage.modules
<class 'sage.modules.free_module_element.FreeModuleElement'>
"""
self._parent = parent
self._is_immutable = is_immutable
def set_immutable(self):
"""
Make this vector immutable. This operation can't be undone.
EXAMPLES::
sage: # needs sage.modules
sage: v = vector([1..5]); v
(1, 2, 3, 4, 5)
sage: v[1] = 10
sage: v.set_immutable()
sage: v[1] = 10
Traceback (most recent call last):
...
ValueError: vector is immutable; please change a copy instead (use copy())
"""
self._is_immutable = 1
cpdef bint is_mutable(self) noexcept:
"""
Return ``True`` if this vector is mutable, i.e., the entries can be
changed.
EXAMPLES::
sage: v = vector(QQ['x,y'], [1..5]); v.is_mutable() # needs sage.modules
True
sage: v.set_immutable() # needs sage.modules
sage: v.is_mutable() # needs sage.modules
False
"""
return not self._is_immutable
cpdef bint is_immutable(self) noexcept:
"""
Return ``True`` if this vector is immutable, i.e., the entries cannot
be changed.
EXAMPLES::
sage: v = vector(QQ['x,y'], [1..5]); v.is_immutable() # needs sage.modules
False
sage: v.set_immutable() # needs sage.modules
sage: v.is_immutable() # needs sage.modules
True
"""
return self._is_immutable
def is_MonoidElement(x):
"""
Return ``True`` if x is of type MonoidElement.
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_MonoidElement is deprecated; use 'isinstance(..., MonoidElement)' instead.")
return isinstance(x, MonoidElement)
cdef class MonoidElement(Element):
"""
Generic element of a monoid.
"""
def order(self):
"""
Return the multiplicative order of ``self``.
"""
return self.multiplicative_order()
def multiplicative_order(self):
"""
Return the multiplicative order of ``self``.
"""
raise NotImplementedError
cpdef _pow_int(self, n):
"""
Return the (integral) power of ``self``.
"""
return arith_generic_power(self, n)
def powers(self, n):
r"""
Return the list `[x^0, x^1, \ldots, x^{n-1}]`.
EXAMPLES::
sage: G = SymmetricGroup(4) # needs sage.groups
sage: g = G([2, 3, 4, 1]) # needs sage.groups
sage: g.powers(4) # needs sage.groups
[(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)]
"""
if n < 0:
raise ValueError("negative number of powers requested")
elif n == 0:
return []
x = self._parent.one()
l = [x]
for i in range(n - 1):
x = x * self
l.append(x)
return l
def __bool__(self):
return True
def is_AdditiveGroupElement(x):
"""
Return ``True`` if x is of type AdditiveGroupElement.
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_AdditiveGroupElement is deprecated; use 'isinstance(..., AdditiveGroupElement)' instead.")
return isinstance(x, AdditiveGroupElement)
cdef class AdditiveGroupElement(ModuleElement):
"""
Generic element of an additive group.
"""
def order(self):
"""
Return additive order of element
"""
return self.additive_order()
def __invert__(self):
raise NotImplementedError("multiplicative inverse not defined for additive group elements")
def is_MultiplicativeGroupElement(x):
"""
Return ``True`` if x is of type MultiplicativeGroupElement.
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_MultiplicativeGroupElement is deprecated; use 'isinstance(..., MultiplicativeGroupElement)' instead.")
return isinstance(x, MultiplicativeGroupElement)
cdef class MultiplicativeGroupElement(MonoidElement):
"""
Generic element of a multiplicative group.
"""
def order(self):
"""
Return the multiplicative order of ``self``.
"""
return self.multiplicative_order()
cpdef _div_(self, right):
"""
Default implementation of division using multiplication by
the inverse.
"""
return self * ~right
def __invert__(self):
r"""
Return the multiplicative inverse of ``self``.
This may cause infinite recursion because of the default definition
of division using inversion in ``_div_``.
"""
return self._parent.one() / self
def is_RingElement(x):
"""
Return ``True`` if x is of type RingElement.
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_RingElement is deprecated; use 'isinstance(..., RingElement)' instead.")
return isinstance(x, RingElement)
cdef class RingElement(ModuleElement):
cpdef _mul_(self, other):
"""
Abstract multiplication method.
TESTS::
sage: from sage.structure.element import RingElement
sage: e = RingElement(Parent())
sage: e * e
Traceback (most recent call last):
...
NotImplementedError: multiplication not implemented for <sage.structure.parent.Parent object at ...>
"""
raise NotImplementedError(f"multiplication not implemented for {self._parent}")
def is_one(self):
return self == self._parent.one()
cpdef _pow_int(self, n):
"""
Return the (integral) power of ``self``.
EXAMPLES::
sage: a = GF(389)['x']['y'](37)
sage: p = sage.structure.element.RingElement.__pow__
sage: p(a,2)
202
sage: p(a,2,1)
Traceback (most recent call last):
...
TypeError: the 3-argument version of pow() is not supported
sage: p(a,388)
1
sage: p(a,2^120)
81
sage: p(a,0)
1
sage: p(a,1) == a
True
sage: p(a,2) * p(a,3) == p(a,5)
True
sage: p(a,3)^2 == p(a,6)
True
sage: p(a,57) * p(a,43) == p(a,100)
True
sage: p(a,-1) == 1/a
True
sage: p(a,200) * p(a,-64) == p(a,136)
True
sage: p(2, 1/2) # needs sage.symbolic
sqrt(2)
TESTS:
These are not testing this code, but they are probably good to have around::
sage: 2r**(SR(2)-1-1r) # needs sage.symbolic
1
sage: 2r^(1/2) # needs sage.symbolic
sqrt(2)
Exponent overflow should throw an :exc:`OverflowError` (:issue:`2956`)::
sage: K.<x,y> = AA[] # needs sage.rings.number_field
sage: x^(2^64 + 12345) # known bug: macos # needs sage.rings.number_field
Traceback (most recent call last):
...
OverflowError: exponent overflow (2147483648)
Another example from :issue:`2956` which always overflows
with Singular 4::
sage: K.<x,y> = ZZ[]
sage: (x^123456)^654321
Traceback (most recent call last):
...
OverflowError: exponent overflow (...)
"""
return arith_generic_power(self, n)
def powers(self, n):
r"""
Return the list `[x^0, x^1, \ldots, x^{n-1}]`.
EXAMPLES::
sage: 5.powers(3)
[1, 5, 25]
"""
if n < 0:
raise ValueError("negative number of powers requested")
elif n == 0:
return []
x = self._parent.one()
l = [x]
for i in range(n - 1):
x = x * self
l.append(x)
return l
cpdef _div_(self, other):
"""
Default implementation of division using the fraction field.
"""
try:
frac = self._parent.fraction_field()
except AttributeError:
raise bin_op_exception('/', self, other)
return frac(self, other)
def __divmod__(self, other):
"""
Return the quotient and remainder of ``self`` divided by ``other``.
This operation may not be defined in all rings.
EXAMPLES::
sage: divmod(5,3)
(1, 2)
sage: divmod(25r,12)
(2, 1)
sage: divmod(25,12r)
(2, 1)
::
sage: R.<x> = QQ[]
sage: f = -19/13*x^5 - x^4 - 2/3*x^3 + 6*x^2 - 2
sage: g = 3*x^2 + 5
sage: q,r = divmod(f,g)
sage: q
-19/39*x^3 - 1/3*x^2 + 23/39*x + 23/9
sage: r
-115/39*x - 133/9
sage: f == q*g + r
True
::
sage: R.<x> = ZZ[]
sage: f = -2*x^5 + x^4 - 9*x^3 - 5*x^2 + 7*x + 4
sage: g = x^2 + 5
sage: q,r = divmod(f,g)
sage: q
-2*x^3 + x^2 + x - 10
sage: r
2*x + 54
sage: f == q*g + r
True
sage: h = 3*x^2 + 5
sage: q,r = divmod(f,h)
sage: q
-3*x - 2
sage: r
-2*x^5 + x^4 + x^2 + 22*x + 14
sage: f == q*h + r
True
::
sage: R.<x> = GF(7)[]
sage: divmod(x^2, x - 1)
(x + 1, 1)
::
sage: divmod(22./7, RR(pi)) # needs sage.symbolic
(1.00040249943477, 0.000000000000000)
"""
try:
return self.quo_rem(other)
except (AttributeError, NotImplementedError):
pass
return (self // other, self % other)
def __invert__(self):
return self._parent.one() / self
def additive_order(self):
"""
Return the additive order of ``self``.
"""
raise NotImplementedError
def multiplicative_order(self):
r"""
Return the multiplicative order of ``self``, if ``self`` is a unit.
This raises an :class:`ArithmeticError` otherwise.
"""
if not self.is_unit():
raise ArithmeticError("self (=%s) must be a unit to have a multiplicative order.")
raise NotImplementedError
def is_nilpotent(self):
"""
Return ``True`` if ``self`` is nilpotent, i.e., some power of ``self``
is 0.
TESTS::
sage: a = QQ(2)
sage: a.is_nilpotent()
False
sage: a = QQ(0)
sage: a.is_nilpotent()
True
sage: m = matrix(QQ, 3, [[3,2,3], [9,0,3], [-9,0,-3]]) # needs sage.modules
sage: m.is_nilpotent() # needs sage.modules
True
"""
if self.is_unit():
return False
if self.is_zero():
return True
raise NotImplementedError
def abs(self):
"""
Return the absolute value of ``self``. (This just calls the ``__abs__``
method, so it is equivalent to the ``abs()`` built-in function.)
EXAMPLES::
sage: RR(-1).abs() # needs sage.rings.real_mpfr
1.00000000000000
sage: ZZ(-1).abs()
1
sage: CC(I).abs() # needs sage.rings.real_mpfr sage.symbolic
1.00000000000000
sage: Mod(-15, 37).abs()
Traceback (most recent call last):
...
ArithmeticError: absolute value not defined on integers modulo n.
"""
return abs(self)
def is_prime(self):
"""
Check whether ``self`` is a prime element.
A *prime* element is a nonzero, non-unit element `p` such that,
whenever `p` divides `ab` for some `a` and `b`, then `p`
divides `a` or `p` divides `b`.
EXAMPLES:
For polynomial rings, prime is the same as irreducible::
sage: # needs sage.libs.singular
sage: R.<x,y> = QQ[]
sage: x.is_prime()
True
sage: (x^2 + y^3).is_prime()
True
sage: (x^2 - y^2).is_prime()
False
sage: R(0).is_prime()
False
sage: R(2).is_prime()
False
For the Gaussian integers::
sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: ZI = K.ring_of_integers()
sage: ZI(3).is_prime()
True
sage: ZI(5).is_prime()
False
sage: ZI(2 + i).is_prime()
True
sage: ZI(0).is_prime()
False
sage: ZI(1).is_prime()
False
In fields, an element is never prime::
sage: RR(0).is_prime()
False
sage: RR(2).is_prime()
False
For integers, :meth:`is_prime` redefines prime numbers to be
positive::
sage: (-2).is_prime()
False
sage: RingElement.is_prime(-2) # needs sage.libs.pari
True
Similarly,
:class:`~sage.rings.number_field.number_field_base.NumberField`
redefines :meth:`is_prime` to determine primality in the ring
of integers::
sage: # needs sage.rings.number_field
sage: (1 + i).is_prime()
True
sage: K(5).is_prime()
False
sage: K(7).is_prime()
True
sage: K(7/13).is_prime()
False
However, for rationals, :meth:`is_prime` *does* follow the
general definition of prime elements in a ring (i.e., always
returns ``False``) since the rationals are not a
:class:`~sage.rings.number_field.number_field_base.NumberField`
in Sage::
sage: QQ(7).is_prime()
False
"""
if not self:
return False
return self._parent.ideal(self).is_prime()
def is_CommutativeRingElement(x):
"""
Return ``True`` if x is of type CommutativeRingElement.
TESTS::
sage: from sage.structure.element import is_CommutativeRingElement
sage: is_CommutativeRingElement(oo)
doctest:warning...
DeprecationWarning: The function is_CommutativeRingElement is deprecated; use 'isinstance(..., CommutativeRingElement)' instead.
See https://github.com/sagemath/sage/issues/38077 for details.
False
sage: is_CommutativeRingElement(1)
True
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_CommutativeRingElement is deprecated; use 'isinstance(..., CommutativeRingElement)' instead.")
return isinstance(x, CommutativeRingElement)
cdef class CommutativeRingElement(RingElement):
"""
Base class for elements of commutative rings.
"""
def inverse_mod(self, I):
r"""
Return an inverse of ``self`` modulo the ideal `I`, if defined,
i.e., if `I` and ``self`` together generate the unit ideal.
EXAMPLES::
sage: # needs sage.rings.finite_rings
sage: F = GF(25)
sage: x = F.gen()
sage: z = F.zero()
sage: x.inverse_mod(F.ideal(z))
2*z2 + 3
sage: x.inverse_mod(F.ideal(1))
1
sage: z.inverse_mod(F.ideal(1))
1
sage: z.inverse_mod(F.ideal(z))
Traceback (most recent call last):
...
ValueError: an element of a proper ideal does not have an inverse modulo that ideal
"""
if I.is_one():
return self.parent().one()
elif self in I:
raise ValueError("an element of a proper ideal does not have an inverse modulo that ideal")
elif hasattr(self, "is_unit") and self.is_unit():
return self.inverse_of_unit()
else:
raise NotImplementedError
def divides(self, x):
"""
Return ``True`` if ``self`` divides x.
EXAMPLES::
sage: P.<x> = PolynomialRing(QQ)
sage: x.divides(x^2)
True
sage: x.divides(x^2 + 2)
False
sage: (x^2 + 2).divides(x)
False
sage: P.<x> = PolynomialRing(ZZ)
sage: x.divides(x^2)
True
sage: x.divides(x^2 + 2)
False
sage: (x^2 + 2).divides(x)
False
:issue:`5347` has been fixed::
sage: K = GF(7)
sage: K(3).divides(1)
True
sage: K(3).divides(K(1))
True
::
sage: R = Integers(128)
sage: R(0).divides(1)
False
sage: R(0).divides(0)
True
sage: R(0).divides(R(0))
True
sage: R(1).divides(0)
True
sage: R(121).divides(R(120))
True
sage: R(120).divides(R(121))
False
If ``x`` has different parent than ``self``, they are first coerced to a
common parent if possible. If this coercion fails, it returns a
:exc:`TypeError`. This fixes :issue:`5759`. ::
sage: Zmod(2)(0).divides(Zmod(2)(0))
True
sage: Zmod(2)(0).divides(Zmod(2)(1))
False
sage: Zmod(5)(1).divides(Zmod(2)(1))
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents:
'Ring of integers modulo 5' and 'Ring of integers modulo 2'
sage: Zmod(35)(4).divides(Zmod(7)(1))
True
sage: Zmod(35)(7).divides(Zmod(7)(1))
False
"""
if have_same_parent(self, x):
try:
if x.is_zero():
return True
except (AttributeError, NotImplementedError):
pass
try:
if self.is_zero():
return False
except (AttributeError, NotImplementedError):
pass
try:
if self.is_unit():
return True
except (AttributeError, NotImplementedError):
pass
try:
if self.is_one():
return True
except (AttributeError, NotImplementedError):
pass
try:
return (x % self).is_zero()
except (TypeError, NotImplementedError):
pass
raise NotImplementedError
else:
a, b = coercion_model.canonical_coercion(self, x)
return a.divides(b)
def mod(self, I):
r"""
Return a representative for ``self`` modulo the ideal I (or the ideal
generated by the elements of I if I is not an ideal.)
EXAMPLES: Integers
Reduction of 5 modulo an ideal::
sage: n = 5
sage: n.mod(3*ZZ)
2
Reduction of 5 modulo the ideal generated by 3::
sage: n.mod(3)
2
Reduction of 5 modulo the ideal generated by 15 and 6, which is `(3)`.
::
sage: n.mod([15,6])
2
EXAMPLES: Univariate polynomials
::
sage: R.<x> = PolynomialRing(QQ)
sage: f = x^3 + x + 1
sage: f.mod(x + 1)
-1
Reduction for `\ZZ[x]`::
sage: R.<x> = PolynomialRing(ZZ)
sage: f = x^3 + x + 1
sage: f.mod(x + 1)
-1
When little is implemented about a given ring, then ``mod`` may
simply return `f`.
EXAMPLES: Multivariate polynomials
We reduce a polynomial in two variables modulo a polynomial
and an ideal::
sage: R.<x,y,z> = PolynomialRing(QQ, 3)
sage: (x^2 + y^2 + z^2).mod(x + y + z) # needs sage.libs.singular
2*y^2 + 2*y*z + 2*z^2
Notice above that `x` is eliminated. In the next example,
both `y` and `z` are eliminated::
sage: (x^2 + y^2 + z^2).mod( (x - y, y - z) ) # needs sage.libs.singular
3*z^2
sage: f = (x^2 + y^2 + z^2)^2; f
x^4 + 2*x^2*y^2 + y^4 + 2*x^2*z^2 + 2*y^2*z^2 + z^4
sage: f.mod( (x - y, y - z) ) # needs sage.libs.singular
9*z^4
In this example `y` is eliminated::
sage: (x^2 + y^2 + z^2).mod( (x^3, y - z) ) # needs sage.libs.singular
x^2 + 2*z^2
"""
from sage.rings.ideal import Ideal_generic
if not isinstance(I, Ideal_generic) or not I.ring() is self._parent:
I = self._parent.ideal(I)
return I.reduce(self)
cdef class Expression(CommutativeRingElement):
r"""
Abstract base class for :class:`~sage.symbolic.expression.Expression`.
This class is defined for the purpose of :func:`isinstance` tests. It should not be
instantiated.
EXAMPLES::
sage: isinstance(SR.var('y'), sage.structure.element.Expression) # needs sage.symbolic
True
By design, there is a unique direct subclass::
sage: len(sage.structure.element.Expression.__subclasses__()) <= 1
True
"""
pass
cdef class Vector(ModuleElementWithMutability):
cdef bint is_sparse_c(self) noexcept:
raise NotImplementedError
cdef bint is_dense_c(self) noexcept:
raise NotImplementedError
def __mul__(left, right):
"""
Multiplication of vector by vector, matrix, or scalar.
AUTHOR:
- Gonzalo Tornaria (2007-06-21) - write test cases and fix them
.. NOTE::
scalar * vector is implemented (and tested) in class RingElement
matrix * vector is implemented (and tested) in class Matrix
TESTS:
Here we test (vector * vector) multiplication::
sage: # needs sage.modules
sage: parent(vector(ZZ, [1,2]) * vector(ZZ, [1,2]))
Integer Ring
sage: parent(vector(ZZ, [1,2]) * vector(QQ, [1,2]))
Rational Field
sage: parent(vector(QQ, [1,2]) * vector(ZZ, [1,2]))
Rational Field
sage: parent(vector(QQ, [1,2]) * vector(QQ, [1,2]))
Rational Field
sage: parent(vector(QQ, [1,2,3,4]) * vector(ZZ['x'], [1,2,3,4])) # needs sage.modules
Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x'], [1,2,3,4]) * vector(QQ, [1,2,3,4])) # needs sage.modules
Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(QQ, [1,2,3,4]) * vector(ZZ['x']['y'], [1,2,3,4])) # needs sage.modules
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x']['y'], [1,2,3,4]) * vector(QQ, [1,2,3,4])) # needs sage.modules
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(QQ['x'], [1,2,3,4]) * vector(ZZ['x']['y'], [1,2,3,4])) # needs sage.modules
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x']['y'], [1,2,3,4]) * vector(QQ['x'], [1,2,3,4])) # needs sage.modules
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(QQ['y'], [1,2,3,4]) * vector(ZZ['x']['y'], [1,2,3,4])) # needs sage.modules
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x']['y'], [1,2,3,4]) * vector(QQ['y'], [1,2,3,4])) # needs sage.modules
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: # needs sage.modules
sage: parent(vector(ZZ['x'], [1,2,3,4]) * vector(ZZ['y'], [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in x over Integer Ring' and
'Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Integer Ring'
sage: parent(vector(ZZ['x'], [1,2,3,4]) * vector(QQ['y'], [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in x over Integer Ring' and
'Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in y over Rational Field'
sage: parent(vector(QQ['x'], [1,2,3,4]) * vector(ZZ['y'], [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field' and
'Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Integer Ring'
sage: parent(vector(QQ['x'], [1,2,3,4]) * vector(QQ['y'], [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field' and
'Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in y over Rational Field'
Here we test (vector * matrix) multiplication::
sage: # needs sage.modules
sage: parent(vector(ZZ, [1,2]) * matrix(ZZ, 2, 2, [1,2,3,4]))
Ambient free module of rank 2 over the principal ideal domain Integer Ring
sage: parent(vector(QQ, [1,2]) * matrix(ZZ, 2, 2, [1,2,3,4]))
Vector space of dimension 2 over Rational Field
sage: parent(vector(ZZ, [1,2]) * matrix(QQ, 2, 2, [1,2,3,4]))
Vector space of dimension 2 over Rational Field
sage: parent(vector(QQ, [1,2]) * matrix(QQ, 2, 2, [1,2,3,4]))
Vector space of dimension 2 over Rational Field
sage: parent(vector(QQ, [1,2]) * matrix(ZZ['x'], 2, 2, [1,2,3,4])) # needs sage.modules
Ambient free module of rank 2
over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x'], [1,2]) * matrix(QQ, 2, 2, [1,2,3,4])) # needs sage.modules
Ambient free module of rank 2
over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(QQ, [1,2]) * matrix(ZZ['x']['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x']['y'], [1,2]) * matrix(QQ, 2, 2, [1,2,3,4])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(QQ['x'], [1,2]) * matrix(ZZ['x']['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x']['y'], [1,2]) * matrix(QQ['x'], 2, 2, [1,2,3,4])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(QQ['y'], [1,2]) * matrix(ZZ['x']['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x']['y'], [1,2]) * matrix(QQ['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: # needs sage.modules
sage: parent(vector(ZZ['x'], [1,2]) * matrix(ZZ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 2 over the integral domain Univariate Polynomial Ring in x over Integer Ring' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Integer Ring'
sage: parent(vector(ZZ['x'], [1,2]) * matrix(QQ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 2 over the integral domain Univariate Polynomial Ring in x over Integer Ring' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Rational Field'
sage: parent(vector(QQ['x'], [1,2]) * matrix(ZZ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Integer Ring'
sage: parent(vector(QQ['x'], [1,2]) * matrix(QQ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Rational Field'
Here we test (vector * scalar) multiplication::
sage: # needs sage.modules
sage: parent(vector(ZZ, [1,2]) * ZZ(1))
Ambient free module of rank 2 over the principal ideal domain Integer Ring
sage: parent(vector(QQ, [1,2]) * ZZ(1))
Vector space of dimension 2 over Rational Field
sage: parent(vector(ZZ, [1,2]) * QQ(1))
Vector space of dimension 2 over Rational Field
sage: parent(vector(QQ, [1,2]) * QQ(1))
Vector space of dimension 2 over Rational Field
sage: parent(vector(QQ, [1,2]) * ZZ['x'](1)) # needs sage.modules
Ambient free module of rank 2
over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x'], [1,2]) * QQ(1)) # needs sage.modules
Ambient free module of rank 2
over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(QQ, [1,2]) * ZZ['x']['y'](1)) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x']['y'], [1,2]) * QQ(1)) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(QQ['x'], [1,2]) * ZZ['x']['y'](1)) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x']['y'], [1,2]) * QQ['x'](1)) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(QQ['y'], [1,2]) * ZZ['x']['y'](1)) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ['x']['y'], [1,2]) * QQ['y'](1)) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: # needs sage.modules
sage: parent(vector(ZZ['x'], [1,2]) * ZZ['y'](1))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 2 over the integral domain Univariate Polynomial Ring in x over Integer Ring' and
'Univariate Polynomial Ring in y over Integer Ring'
sage: parent(vector(ZZ['x'], [1,2]) * QQ['y'](1))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 2 over the integral domain Univariate Polynomial Ring in x over Integer Ring' and
'Univariate Polynomial Ring in y over Rational Field'
sage: parent(vector(QQ['x'], [1,2]) * ZZ['y'](1))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field' and
'Univariate Polynomial Ring in y over Integer Ring'
sage: parent(vector(QQ['x'], [1,2]) * QQ['y'](1))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field' and
'Univariate Polynomial Ring in y over Rational Field'
Here we test (scalar * vector) multiplication::
sage: # needs sage.modules
sage: parent(ZZ(1) * vector(ZZ, [1,2]))
Ambient free module of rank 2 over the principal ideal domain Integer Ring
sage: parent(QQ(1) * vector(ZZ, [1,2]))
Vector space of dimension 2 over Rational Field
sage: parent(ZZ(1) * vector(QQ, [1,2]))
Vector space of dimension 2 over Rational Field
sage: parent(QQ(1) * vector(QQ, [1,2]))
Vector space of dimension 2 over Rational Field
sage: parent(QQ(1) * vector(ZZ['x'], [1,2])) # needs sage.modules
Ambient free module of rank 2
over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
sage: parent(ZZ['x'](1) * vector(QQ, [1,2])) # needs sage.modules
Ambient free module of rank 2
over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
sage: parent(QQ(1) * vector(ZZ['x']['y'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(ZZ['x']['y'](1) * vector(QQ, [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(QQ['x'](1) * vector(ZZ['x']['y'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(ZZ['x']['y'](1) * vector(QQ['x'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(QQ['y'](1) * vector(ZZ['x']['y'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(ZZ['x']['y'](1) * vector(QQ['y'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: # needs sage.modules
sage: parent(ZZ['x'](1) * vector(ZZ['y'], [1,2]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Univariate Polynomial Ring in x over Integer Ring' and
'Ambient free module of rank 2 over the integral domain Univariate Polynomial Ring in y over Integer Ring'
sage: parent(ZZ['x'](1) * vector(QQ['y'], [1,2]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Univariate Polynomial Ring in x over Integer Ring' and
'Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in y over Rational Field'
sage: parent(QQ['x'](1) * vector(ZZ['y'], [1,2]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Univariate Polynomial Ring in x over Rational Field' and
'Ambient free module of rank 2 over the integral domain Univariate Polynomial Ring in y over Integer Ring'
sage: parent(QQ['x'](1) * vector(QQ['y'], [1,2]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Univariate Polynomial Ring in x over Rational Field' and
'Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in y over Rational Field'
"""
if have_same_parent(left, right):
return (<Vector>left)._dot_product_(<Vector>right)
return coercion_model.bin_op(left, right, mul)
cpdef _dot_product_(Vector left, Vector right):
return left._dot_product_coerce_(right)
cpdef _dot_product_coerce_(Vector left, Vector right):
raise bin_op_exception('*', left, right)
cpdef _pairwise_product_(Vector left, Vector right):
raise TypeError("unsupported operation for '%s' and '%s'" % (parent(left), parent(right)))
def __truediv__(self, right):
"""
Divide this vector by a scalar, vector or matrix.
TESTS::
sage: # needs sage.modules
sage: A = matrix([[1, 2], [0, 3]])
sage: b = vector([0, 1])
sage: x = b / A; x
(0, 1/3)
sage: x == b * ~A
True
sage: A = matrix([[1, 2], [0, 3], [1, 5]])
sage: (b / A) * A == b
True
"""
right = py_scalar_to_element(right)
if isinstance(right, RingElement):
return self * ~right
if isinstance(right, Vector):
try:
W = (<Vector>right)._parent.submodule([right])
return W.coordinates(self)[0] / W.coordinates(right)[0]
except ArithmeticError:
if right.is_zero():
raise ZeroDivisionError("division by zero vector")
else:
raise ArithmeticError("vector is not in free module")
if isinstance(right, Matrix):
return right.solve_left(self)
raise bin_op_exception('/', self, right)
def _magma_init_(self, magma):
"""
Return string that evaluates in Magma to something equivalent
to this vector.
EXAMPLES::
sage: # optional - magma, needs sage.modules
sage: v = vector([1,2,3])
sage: v._magma_init_(magma)
'_sage_[...]![1,2,3]'
sage: mv = magma(v); mv
(1 2 3)
sage: mv.Type()
ModTupRngElt
sage: mv.Parent()
Full RSpace of degree 3 over Integer Ring
sage: # optional - magma, needs sage.modules
sage: v = vector(QQ, [1/2, 3/4, 5/6])
sage: mv = magma(v); mv
(1/2 3/4 5/6)
sage: mv.Type()
ModTupFldElt
sage: mv.Parent()
Full Vector space of degree 3 over Rational Field
A more demanding example::
sage: # optional - magma, needs sage.modules
sage: R.<x,y,z> = QQ[]
sage: v = vector([x^3, y, 2/3*z + x/y])
sage: magma(v)
( x^3 y (2/3*y*z + x)/y)
sage: magma(v).Parent()
Full Vector space of degree 3
over Multivariate rational function field of rank 3 over Rational Field
"""
V = magma(self._parent)
v = [x._magma_init_(magma) for x in self.list()]
return '%s![%s]' % (V.name(), ','.join(v))
def is_Vector(x):
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_Vector is deprecated; use 'isinstance(..., Vector)' instead.")
return isinstance(x, Vector)
cdef class Matrix(ModuleElement):
cdef bint is_sparse_c(self) noexcept:
raise NotImplementedError
cdef bint is_dense_c(self) noexcept:
raise NotImplementedError
def __mul__(left, right):
"""
Multiplication of matrix by matrix, vector, or scalar.
AUTHOR:
- Gonzalo Tornaria (2007-06-25) - write test cases and fix them
.. NOTE::
scalar * matrix is implemented (and tested) in class RingElement
vector * matrix is implemented (and tested) in class Vector
TESTS:
Here we test (matrix * matrix) multiplication::
sage: # needs sage.modules
sage: parent(matrix(ZZ, 2, 2, [1,2,3,4]) * matrix(ZZ, 2, 2, [1,2,3,4]))
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * matrix(ZZ, 2, 2, [1,2,3,4]))
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: parent(matrix(ZZ, 2, 2, [1,2,3,4]) * matrix(QQ, 2, 2, [1,2,3,4]))
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * matrix(QQ, 2, 2, [1,2,3,4]))
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * matrix(ZZ['x'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x'], 2, 2, [1,2,3,4]) * matrix(QQ, 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * matrix(ZZ['x']['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x']['y'], 2, 2, [1,2,3,4]) * matrix(QQ, 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(QQ['x'], 2, 2, [1,2,3,4]) * matrix(ZZ['x']['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x']['y'], 2, 2, [1,2,3,4]) * matrix(QQ['x'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(QQ['y'], 2, 2, [1,2,3,4]) * matrix(ZZ['x']['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x']['y'], 2, 2, [1,2,3,4]) * matrix(QQ['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: # needs sage.modules
sage: parent(matrix(ZZ['x'], 2, 2, [1,2,3,4]) * matrix(ZZ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Integer Ring' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Integer Ring'
sage: parent(matrix(ZZ['x'], 2, 2, [1,2,3,4]) * matrix(QQ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Integer Ring' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Rational Field'
sage: parent(matrix(QQ['x'], 2, 2, [1,2,3,4]) * matrix(ZZ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Integer Ring'
sage: parent(matrix(QQ['x'], 2, 2, [1,2,3,4]) * matrix(QQ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Rational Field'
We test that the bug reported in :issue:`27352` has been fixed::
sage: A = matrix(QQ, [[1, 2], [-1, 0], [1, 1]]) # needs sage.modules
sage: B = matrix(QQ, [[0, 4], [1, -1], [1, 2]]) # needs sage.modules
sage: A * B # needs sage.modules
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 3 by 2 dense matrices over Rational Field' and 'Full MatrixSpace of 3 by 2 dense matrices over Rational Field'
Here we test (matrix * vector) multiplication::
sage: # needs sage.modules
sage: parent(matrix(ZZ, 2, 2, [1,2,3,4]) * vector(ZZ, [1,2]))
Ambient free module of rank 2 over the principal ideal domain Integer Ring
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * vector(ZZ, [1,2]))
Vector space of dimension 2 over Rational Field
sage: parent(matrix(ZZ, 2, 2, [1,2,3,4]) * vector(QQ, [1,2]))
Vector space of dimension 2 over Rational Field
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * vector(QQ, [1,2]))
Vector space of dimension 2 over Rational Field
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * vector(ZZ['x'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the principal ideal domain
Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x'], 2, 2, [1,2,3,4]) * vector(QQ, [1,2])) # needs sage.modules
Ambient free module of rank 2 over the principal ideal domain
Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * vector(ZZ['x']['y'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x']['y'], 2, 2, [1,2,3,4]) * vector(QQ, [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(QQ['x'], 2, 2, [1,2,3,4]) * vector(ZZ['x']['y'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x']['y'], 2, 2, [1,2,3,4]) * vector(QQ['x'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(QQ['y'], 2, 2, [1,2,3,4]) * vector(ZZ['x']['y'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x']['y'], 2, 2, [1,2,3,4]) * vector(QQ['y'], [1,2])) # needs sage.modules
Ambient free module of rank 2 over the integral domain
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: # needs sage.modules
sage: parent(matrix(ZZ['x'], 2, 2, [1,2,3,4]) * vector(ZZ['y'], [1,2]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Integer Ring' and
'Ambient free module of rank 2 over the integral domain Univariate Polynomial Ring in y over Integer Ring'
sage: parent(matrix(ZZ['x'], 2, 2, [1,2,3,4]) * vector(QQ['y'], [1,2]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Integer Ring' and
'Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in y over Rational Field'
sage: parent(matrix(QQ['x'], 2, 2, [1,2,3,4]) * vector(ZZ['y'], [1,2]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field' and
'Ambient free module of rank 2 over the integral domain Univariate Polynomial Ring in y over Integer Ring'
sage: parent(matrix(QQ['x'], 2, 2, [1,2,3,4]) * vector(QQ['y'], [1,2]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field' and
'Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in y over Rational Field'
Here we test (matrix * scalar) multiplication::
sage: # needs sage.modules
sage: parent(matrix(ZZ, 2, 2, [1,2,3,4]) * ZZ(1))
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * ZZ(1))
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: parent(matrix(ZZ, 2, 2, [1,2,3,4]) * QQ(1))
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * QQ(1))
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * ZZ['x'](1)) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x'], 2, 2, [1,2,3,4]) * QQ(1)) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(QQ, 2, 2, [1,2,3,4]) * ZZ['x']['y'](1)) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x']['y'], 2, 2, [1,2,3,4]) * QQ(1)) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(QQ['x'], 2, 2, [1,2,3,4]) * ZZ['x']['y'](1)) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x']['y'], 2, 2, [1,2,3,4]) * QQ['x'](1)) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(QQ['y'], 2, 2, [1,2,3,4]) * ZZ['x']['y'](1)) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(matrix(ZZ['x']['y'], 2, 2, [1,2,3,4]) * QQ['y'](1)) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: # needs sage.modules
sage: parent(matrix(ZZ['x'], 2, 2, [1,2,3,4]) * ZZ['y'](1))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Integer Ring' and
'Univariate Polynomial Ring in y over Integer Ring'
sage: parent(matrix(ZZ['x'], 2, 2, [1,2,3,4]) * QQ['y'](1))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Integer Ring' and
'Univariate Polynomial Ring in y over Rational Field'
sage: parent(matrix(QQ['x'], 2, 2, [1,2,3,4]) * ZZ['y'](1))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field' and
'Univariate Polynomial Ring in y over Integer Ring'
sage: parent(matrix(QQ['x'], 2, 2, [1,2,3,4]) * QQ['y'](1))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field' and
'Univariate Polynomial Ring in y over Rational Field'
Here we test (scalar * matrix) multiplication::
sage: # needs sage.modules
sage: parent(ZZ(1) * matrix(ZZ, 2, 2, [1,2,3,4]))
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: parent(QQ(1) * matrix(ZZ, 2, 2, [1,2,3,4]))
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: parent(ZZ(1) * matrix(QQ, 2, 2, [1,2,3,4]))
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: parent(QQ(1) * matrix(QQ, 2, 2, [1,2,3,4]))
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: parent(QQ(1) * matrix(ZZ['x'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field
sage: parent(ZZ['x'](1) * matrix(QQ, 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational Field
sage: parent(QQ(1) * matrix(ZZ['x']['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(ZZ['x']['y'](1) * matrix(QQ, 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(QQ['x'](1) * matrix(ZZ['x']['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(ZZ['x']['y'](1) * matrix(QQ['x'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices over
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(QQ['y'](1) * matrix(ZZ['x']['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices
over Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: parent(ZZ['x']['y'](1) * matrix(QQ['y'], 2, 2, [1,2,3,4])) # needs sage.modules
Full MatrixSpace of 2 by 2 dense matrices
over Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
sage: # needs sage.modules
sage: parent(ZZ['x'](1) * matrix(ZZ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Univariate Polynomial Ring in x over Integer Ring' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Integer Ring'
sage: parent(ZZ['x'](1) * matrix(QQ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Univariate Polynomial Ring in x over Integer Ring' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Rational Field'
sage: parent(QQ['x'](1) * matrix(ZZ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Univariate Polynomial Ring in x over Rational Field' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Integer Ring'
sage: parent(QQ['x'](1) * matrix(QQ['y'], 2, 2, [1,2,3,4]))
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'Univariate Polynomial Ring in x over Rational Field' and
'Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Rational Field'
Examples with matrices having matrix coefficients::
sage: m = matrix # needs sage.modules
sage: a = m([[m([[1,2],[3,4]]),m([[5,6],[7,8]])],[m([[9,10],[11,12]]),m([[13,14],[15,16]])]]) # needs sage.modules
sage: 3 * a # needs sage.modules
[[ 3 6]
[ 9 12] [15 18]
[21 24]]
[[27 30]
[33 36] [39 42]
[45 48]]
sage: m = matrix # needs sage.modules
sage: a = m([[m([[1,2],[3,4]]),m([[5,6],[7,8]])],[m([[9,10],[11,12]]),m([[13,14],[15,16]])]]) # needs sage.modules
sage: a * 3 # needs sage.modules
[[ 3 6]
[ 9 12] [15 18]
[21 24]]
[[27 30]
[33 36] [39 42]
[45 48]]
"""
cdef int cl = classify_elements(left, right)
if HAVE_SAME_PARENT(cl):
if (<Matrix>left)._nrows == (<Matrix>left)._ncols:
return (<Matrix>left)._matrix_times_matrix_(<Matrix>right)
else:
parent = (<Matrix>left)._parent
raise TypeError("unsupported operand parent(s) for *: '{}' and '{}'".format(parent, parent))
if BOTH_ARE_ELEMENT(cl):
return coercion_model.bin_op(left, right, mul)
cdef long value
cdef int err = -1
try:
integer_check_long_py(right, &value, &err)
if not err:
return (<Element>left)._mul_long(value)
integer_check_long_py(left, &value, &err)
if not err:
return (<Element>right)._mul_long(value)
return coercion_model.bin_op(left, right, mul)
except TypeError:
return NotImplemented
def __truediv__(left, right):
"""
Division of the matrix ``left`` by the matrix or scalar
``right``.
EXAMPLES::
sage: # needs sage.modules
sage: a = matrix(ZZ, 2, range(4))
sage: operator.truediv(a, 5)
[ 0 1/5]
[2/5 3/5]
sage: a = matrix(ZZ, 2, range(4))
sage: b = matrix(ZZ, 2, [1,1,0,5])
sage: operator.truediv(a, b)
[ 0 1/5]
[ 2 1/5]
sage: c = matrix(QQ, 2, [3,2,5,7])
sage: operator.truediv(c, a)
[-5/2 3/2]
[-1/2 5/2]
TESTS::
sage: # needs sage.modules
sage: a = matrix(ZZ, [[1, 2], [0, 3]])
sage: b = matrix(ZZ, 3, 2, range(6))
sage: x = b / a; x
[ 0 1/3]
[ 2 -1/3]
[ 4 -1]
sage: x == b * ~a
True
sage: a = matrix(ZZ, [[1, 2], [0, 3], [1, 5]])
sage: (b / a) * a == b
True
"""
if isinstance(right, Matrix):
return right.solve_left(left)
return coercion_model.bin_op(left, right, truediv)
cdef _vector_times_matrix_(matrix_right, Vector vector_left):
raise TypeError
cdef _matrix_times_vector_(matrix_left, Vector vector_right):
raise TypeError
cdef _matrix_times_matrix_(left, Matrix right):
raise TypeError
def is_Matrix(x):
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_Matrix is deprecated; use 'isinstance(..., Matrix)' instead.")
return isinstance(x, Matrix)
def is_IntegralDomainElement(x):
"""
Return ``True`` if x is of type IntegralDomainElement.
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_IntegralDomainElement is deprecated; use 'isinstance(..., IntegralDomainElement)' instead.")
return isinstance(x, IntegralDomainElement)
cdef class IntegralDomainElement(CommutativeRingElement):
def is_nilpotent(self):
return self.is_zero()
def is_DedekindDomainElement(x):
"""
Return ``True`` if x is of type DedekindDomainElement.
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_DedekindDomainElement is deprecated; use 'isinstance(..., DedekindDomainElement)' instead.")
return isinstance(x, DedekindDomainElement)
cdef class DedekindDomainElement(IntegralDomainElement):
pass
def is_PrincipalIdealDomainElement(x):
"""
Return ``True`` if x is of type PrincipalIdealDomainElement.
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_PrincipalIdealDomainElement is deprecated; use 'isinstance(..., PrincipalIdealDomainElement)' instead.")
return isinstance(x, PrincipalIdealDomainElement)
cdef class PrincipalIdealDomainElement(DedekindDomainElement):
def gcd(self, right):
r"""
Return the greatest common divisor of ``self`` and ``other``.
TESTS:
:issue:`30849`::
sage: 2.gcd(pari(3)) # needs sage.libs.pari
1
sage: type(2.gcd(pari(3))) # needs sage.libs.pari
<class 'sage.rings.integer.Integer'>
sage: 2.gcd(pari('1/3')) # needs sage.libs.pari
1/3
sage: type(2.gcd(pari('1/3'))) # needs sage.libs.pari
<class 'sage.rings.rational.Rational'>
sage: import gmpy2
sage: 2.gcd(gmpy2.mpz(3))
1
sage: type(2.gcd(gmpy2.mpz(3)))
<class 'sage.rings.integer.Integer'>
sage: 2.gcd(gmpy2.mpq(1,3))
1/3
sage: type(2.gcd(pari('1/3'))) # needs sage.libs.pari
<class 'sage.rings.rational.Rational'>
"""
if not isinstance(right, Element):
right = py_scalar_to_element(right)
if not isinstance(right, Element):
right = right.sage()
if not ((<Element>right)._parent is self._parent):
from sage.arith.misc import GCD as gcd
return coercion_model.bin_op(self, right, gcd)
return self._gcd(right)
def lcm(self, right):
"""
Return the least common multiple of ``self`` and ``right``.
TESTS:
:issue:`30849`::
sage: 2.lcm(pari(3)) # needs sage.libs.pari
6
sage: type(2.lcm(pari(3))) # needs sage.libs.pari
<class 'sage.rings.integer.Integer'>
sage: 2.lcm(pari('1/3')) # needs sage.libs.pari
2
sage: type(2.lcm(pari('1/3'))) # needs sage.libs.pari
<class 'sage.rings.rational.Rational'>
sage: import gmpy2
sage: 2.lcm(gmpy2.mpz(3))
6
sage: type(2.lcm(gmpy2.mpz(3)))
<class 'sage.rings.integer.Integer'>
"""
if not isinstance(right, Element):
right = py_scalar_to_element(right)
if not isinstance(right, Element):
right = right.sage()
if not ((<Element>right)._parent is self._parent):
from sage.arith.functions import lcm
return coercion_model.bin_op(self, right, lcm)
return self._lcm(right)
PY_SET_TP_NEW(EuclideanDomainElement, Element)
def is_EuclideanDomainElement(x):
"""
Return ``True`` if x is of type EuclideanDomainElement.
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_EuclideanDomainElement is deprecated; use 'isinstance(..., EuclideanDomainElement)' instead.")
return isinstance(x, EuclideanDomainElement)
cdef class EuclideanDomainElement(PrincipalIdealDomainElement):
def degree(self):
raise NotImplementedError
def leading_coefficient(self):
raise NotImplementedError
def quo_rem(self, other):
raise NotImplementedError
cpdef _floordiv_(self, right):
"""
Quotient of division of ``self`` by ``other``. This is denoted ``//``.
This default implementation assumes that ``quo_rem`` has been
implemented.
EXAMPLES::
sage: cython( # needs sage.misc.cython
....: '''
....: from sage.structure.element cimport EuclideanDomainElement
....: cdef class MyElt(EuclideanDomainElement):
....: def quo_rem(self, other):
....: return self._parent.var('quo,rem')
....: ''')
sage: e = MyElt(SR) # needs sage.misc.cython sage.symbolic
sage: e // e # needs sage.misc.cython sage.symbolic
quo
"""
Q, _ = self.quo_rem(right)
return Q
cpdef _mod_(self, other):
"""
Remainder of division of ``self`` by ``other``.
This default implementation assumes that ``quo_rem`` has been
implemented.
EXAMPLES::
sage: R.<x> = ZZ[]
sage: x % (x+1)
-1
sage: (x^3 + x - 1) % (x^2 - 1)
2*x - 1
::
sage: cython( # needs sage.misc.cython
....: '''
....: from sage.structure.element cimport EuclideanDomainElement
....: cdef class MyElt(EuclideanDomainElement):
....: def quo_rem(self, other):
....: return self._parent.var('quo,rem')
....: ''')
sage: e = MyElt(SR) # needs sage.misc.cython sage.symbolic
sage: e % e # needs sage.misc.cython sage.symbolic
rem
"""
_, R = self.quo_rem(other)
return R
def is_FieldElement(x):
"""
Return ``True`` if x is of type FieldElement.
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_FieldElement is deprecated; use 'isinstance(..., FieldElement)' instead.")
return isinstance(x, FieldElement)
cdef class FieldElement(CommutativeRingElement):
cpdef _floordiv_(self, right):
"""
Return the quotient of ``self`` and ``other``. Since these are field
elements, the floor division is exactly the same as usual division.
EXAMPLES::
sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<b> = NumberField(x^4 + x^2 + 2/3)
sage: c = (1+b) // (1-b); c
3/4*b^3 + 3/4*b^2 + 3/2*b + 1/2
sage: (1+b) / (1-b) == c
True
sage: c * (1-b)
b + 1
"""
return self._div_(right)
def is_unit(self):
r"""
Return ``True`` if ``self`` is a unit in its parent ring.
EXAMPLES::
sage: a = 2/3; a.is_unit()
True
On the other hand, 2 is not a unit, since its parent is `\ZZ`.
::
sage: a = 2; a.is_unit()
False
sage: parent(a)
Integer Ring
However, a is a unit when viewed as an element of QQ::
sage: a = QQ(2); a.is_unit()
True
"""
return not not self
def _lcm(self, FieldElement other):
"""
Return the least common multiple of ``self`` and ``other``.
"""
if self.is_zero() and other.is_zero():
return self
else:
return self._parent(1)
def quo_rem(self, right):
r"""
Return the quotient and remainder obtained by dividing ``self`` by
``right``. Since this element lives in a field, the remainder is always
zero and the quotient is ``self/right``.
TESTS:
Test if :issue:`8671` is fixed::
sage: # needs sage.libs.pari sage.libs.singular
sage: R.<x,y> = QQ[]
sage: S.<a,b> = R.quo(y^2 + 1)
sage: S.is_field = lambda: False
sage: F = Frac(S); u = F.one()
sage: u.quo_rem(u)
(1, 0)
"""
if not isinstance(right, FieldElement) or not (parent(right) is self._parent):
right = self.parent()(right)
return self/right, 0
def divides(self, FieldElement other):
r"""
Check whether ``self`` divides ``other``, for field elements.
Since this is a field, all values divide all other values,
except that zero does not divide any nonzero values.
EXAMPLES::
sage: # needs sage.rings.number_field sage.symbolic
sage: K.<rt3> = QQ[sqrt(3)]
sage: K(0).divides(rt3)
False
sage: rt3.divides(K(17))
True
sage: K(0).divides(K(0))
True
sage: rt3.divides(K(0))
True
"""
if not (other._parent is self._parent):
other = self.parent()(other)
return bool(self) or other.is_zero()
def canonical_associate(self):
"""
Return a canonical associate.
EXAMPLES::
sage: R.<x,y>=QQ[]; k=R.fraction_field()
sage: (x/y).canonical_associate()
(1, x/y)
sage: (0).canonical_associate()
(0, 1)
"""
P = self.parent()
if self.is_zero():
return (P.zero(), P.one())
return (P.one(), self)
def is_AlgebraElement(x):
"""
Return ``True`` if x is of type AlgebraElement.
TESTS::
sage: from sage.structure.element import is_AlgebraElement
sage: R.<x,y> = FreeAlgebra(QQ, 2) # needs sage.combinat sage.modules
sage: is_AlgebraElement(x * y) # needs sage.combinat sage.modules
doctest:warning...
DeprecationWarning: The function is_AlgebraElement is deprecated; use 'isinstance(..., AlgebraElement)' instead.
See https://github.com/sagemath/sage/issues/38077 for details.
True
sage: is_AlgebraElement(1)
False
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_AlgebraElement is deprecated; use 'isinstance(..., AlgebraElement)' instead.")
return isinstance(x, AlgebraElement)
cdef class AlgebraElement(RingElement):
pass
def is_CommutativeAlgebraElement(x):
"""
Return ``True`` if x is of type CommutativeAlgebraElement.
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_CommutativeAlgebraElement is deprecated; use 'isinstance(..., CommutativeAlgebraElement)' instead.")
return isinstance(x, CommutativeAlgebraElement)
cdef class CommutativeAlgebraElement(CommutativeRingElement):
pass
def is_InfinityElement(x):
"""
Return ``True`` if x is of type InfinityElement.
TESTS::
sage: from sage.structure.element import is_InfinityElement
sage: is_InfinityElement(1)
doctest:warning...
DeprecationWarning: The function is_InfinityElement is deprecated; use 'isinstance(..., InfinityElement)' instead.
See https://github.com/sagemath/sage/issues/38077 for details.
False
sage: is_InfinityElement(oo)
True
"""
from sage.misc.superseded import deprecation_cython
deprecation_cython(38077, "The function is_InfinityElement is deprecated; use 'isinstance(..., InfinityElement)' instead.")
return isinstance(x, InfinityElement)
cdef class InfinityElement(RingElement):
def __invert__(self):
from sage.rings.integer_ring import ZZ
return ZZ(0)
cpdef canonical_coercion(x, y):
"""
``canonical_coercion(x,y)`` is what is called before doing an
arithmetic operation between ``x`` and ``y``. It returns a pair ``(z,w)``
such that ``z`` is got from ``x`` and ``w`` from ``y`` via canonical coercion and
the parents of ``z`` and ``w`` are identical.
EXAMPLES::
sage: A = Matrix([[0, 1], [1, 0]]) # needs sage.modules
sage: canonical_coercion(A, 1) # needs sage.modules
(
[0 1] [1 0]
[1 0], [0 1]
)
"""
return coercion_model.canonical_coercion(x, y)
cpdef bin_op(x, y, op):
return coercion_model.bin_op(x, y, op)
globals()["coercion_model"] = coercion_model
def get_coercion_model():
"""
Return the global coercion model.
EXAMPLES::
sage: import sage.structure.element as e
sage: cm = e.get_coercion_model()
sage: cm
<sage.structure.coerce.CoercionModel object at ...>
sage: cm is coercion_model
True
"""
return coercion_model
def coercion_traceback(dump=True):
r"""
This function is very helpful in debugging coercion errors. It prints
the tracebacks of all the errors caught in the coercion detection. Note
that failure is cached, so some errors may be omitted the second time
around (as it remembers not to retry failed paths for speed reasons.
For performance and caching reasons, exception recording must be
explicitly enabled before using this function.
EXAMPLES::
sage: cm = sage.structure.element.get_coercion_model()
sage: cm.record_exceptions()
sage: 1 + 1/5
6/5
sage: coercion_traceback() # Should be empty, as all went well.
sage: 1/5 + GF(5).gen()
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +:
'Rational Field' and 'Finite Field of size 5'
sage: coercion_traceback()
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents:
'Rational Field' and 'Finite Field of size 5'
"""
if dump:
for traceback in coercion_model.exception_stack():
print(traceback)
else:
return coercion_model.exception_stack()
def coerce_binop(method):
r"""
Decorator for a binary operator method for applying coercion to the
arguments before calling the method.
Consider a parent class in the category framework, `S`, whose element class
expose a method `binop`. If `a` and `b` are elements of `S`, then
`a.binop(b)` behaves as expected. If `a` and `b` are not elements of `S`,
but rather have a common parent `T` whose element class also exposes
`binop`, we would rather expect `a.binop(b)` to compute `aa.binop(bb)`,
where `aa = T(a)` and `bb = T(b)`. This decorator ensures that behaviour
without having to otherwise modify the implementation of `binop` on the
element class of `A`.
Since coercion will be attempted on the arguments of the decorated method, a
`TypeError` will be thrown if there is no common parent between the
elements. An `AttributeError` or `NotImplementedError` or similar will be
thrown if there is a common parent of the arguments, but its element class
does not implement a method of the same name as the decorated method.
EXAMPLES:
Sparse polynomial rings uses ``@coerce_binop`` on ``gcd``::
sage: S.<x> = PolynomialRing(ZZ, sparse=True)
sage: f = x^2
sage: g = x
sage: f.gcd(g) #indirect doctest
x
sage: T = PolynomialRing(QQ, name='x', sparse=True)
sage: h = 1/2*T(x)
sage: u = f.gcd(h); u #indirect doctest
x
sage: u.parent() == T
True
Another real example::
sage: R1 = QQ['x,y']
sage: R2 = QQ['x,y,z']
sage: f = R1(1)
sage: g = R1(2)
sage: h = R2(1)
sage: f.gcd(g)
1
sage: f.gcd(g, algorithm='modular')
1
sage: f.gcd(h)
1
sage: f.gcd(h, algorithm='modular')
1
sage: h.gcd(f)
1
sage: h.gcd(f, 'modular')
1
We demonstrate a small class using ``@coerce_binop`` on a method::
sage: from sage.structure.element import coerce_binop
sage: class MyRational(Rational):
....: def __init__(self, value):
....: self.v = value
....: @coerce_binop
....: def test_add(self, other, keyword='z'):
....: return (self.v, other, keyword)
Calls func directly if the two arguments have the same parent::
sage: x = MyRational(1)
sage: x.test_add(1/2)
(1, 1/2, 'z')
sage: x.test_add(1/2, keyword=3)
(1, 1/2, 3)
Passes through coercion and does a method lookup if the left operand is not
the same. If the common parent's element class does not have a method of the
same name, an exception is raised::
sage: x.test_add(2)
(1, 2, 'z')
sage: x.test_add(2, keyword=3)
(1, 2, 3)
sage: x.test_add(CC(2))
Traceback (most recent call last):
...
AttributeError: 'sage.rings.complex_mpfr.ComplexNumber' object has no attribute 'test_add'...
TESTS:
Test that additional arguments given to the method do not override
the ``self`` argument, see :issue:`21322`::
sage: f.gcd(g, 1)
Traceback (most recent call last):
...
TypeError: algorithm 1 not supported
"""
@sage_wraps(method)
def new_method(self, other, *args, **kwargs):
if have_same_parent(self, other):
return method(self, other, *args, **kwargs)
else:
a, b = coercion_model.canonical_coercion(self, other)
if a is self:
return method(a, b, *args, **kwargs)
else:
return getattr(a, method.__name__)(b, *args, **kwargs)
return new_method
