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
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
FiniteFieldElement
CommutativeAlgebraElement
AlgebraElement (note -- can't derive from module, since no multiple inheritance)
CommutativeAlgebra ??? (should be removed from element.pxd)
Matrix
InfinityElement
PlusInfinityElement
MinusInfinityElement
AdditiveGroupElement
Vector
MonoidElement
MultiplicativeGroupElement
ElementWithCachedMethod
How to Define a New Element Class
---------------------------------
Elements typically define a method ``_new_c``, e.g.,
::
cdef _new_c(self, defining data):
cdef FreeModuleElement_generic_dense x
x = PY_NEW(FreeModuleElement_generic_dense)
x._parent = self._parent
x._entries = v
that creates a new sibling very quickly from defining data
with assumed properties.
Sage has a special system in place for handling arithmetic operations
for all Element subclasses. There are various rules that must be
followed by both arithmetic implementers and callers.
A quick summary for the impatient:
- To implement addition for any Element class, override def _add_().
- If you want to add x and y, whose parents you know are IDENTICAL,
you may call _add_(x, y). This will be the fastest way to guarantee
that the correct implementation gets called. Of course you can still
always use "x + y".
Now 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 of RingElements as an example; all other operators
and classes are similar. There are four relevant functions.
- **def RingElement.__add__**
This function is called by Python or Pyrex when the binary "+" operator
is encountered. It ASSUMES that at least one of its arguments is a
RingElement; only a really twisted programmer would violate this
condition. It has a fast pathway to deal with the most common case
where the arguments have the same parent. Otherwise, it uses the coercion
module to work out how to make them have the same parent. After any
necessary coercions have been performed, it calls _add_ to dispatch to
the correct underlying addition implementation.
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 RingElement._add_**
This is the function you should override to implement addition in a
python subclass of RingElement.
.. warning::
if you override this in a *Cython* class, it won't get called.
You should override _add_ instead. It is especially important to
keep this in mind whenever you move a class down from Python to
Cython.
The two arguments to this function are guaranteed to have the
SAME PARENT. Its return value MUST have the SAME PARENT as its
arguments.
If you want to add two objects from python, and you know that their
parents are the same object, you are encouraged to call this function
directly, instead of using "x + y".
The default implementation of this function is to call _add_,
so if no-one has defined a python implementation, the correct Pyrex
implementation will get called.
- **cpdef RingElement._add_**
This is the function you should override to implement addition in a
Pyrex subclass of RingElement.
The two arguments to this function are guaranteed to have the
SAME PARENT. Its return value MUST have the SAME PARENT as its
arguments.
The default implementation of this function is to raise a
NotImplementedError, which will happen if no-one has supplied
implementations of either _add_.
For speed, there are also **inplace** version of the arithmetic commands.
DD NOT call them directly, they may mutate the object and will be called
when and only when it has been determined that the old object will no longer
be accessible from the calling function after this operation.
- **def RingElement._iadd_**
This is the function you should override to inplace implement addition
in a Python subclass of RingElement.
The two arguments to this function are guaranteed to have the
SAME PARENT. Its return value MUST have the SAME PARENT as its
arguments.
The default implementation of this function is to call _add_,
so if no-one has defined a Python implementation, the correct Cython
implementation will get called.
"""
cdef extern from *:
ctypedef struct RefPyObject "PyObject":
int ob_refcnt
include "../ext/cdefs.pxi"
include "../ext/stdsage.pxi"
include "../ext/python.pxi"
import operator, types
import sys
import traceback
import sage.misc.sageinspect as sageinspect
cdef MethodType
from types import MethodType
from sage.categories.category import Category
from sage.structure.parent cimport Parent, AttributeErrorMessage
from sage.structure.parent import is_extension_type, getattr_from_other_class
from sage.misc.lazy_format import LazyFormat
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)
def py_scalar_to_element(py):
from sage.rings.integer_ring import ZZ
from sage.rings.real_double import RDF
from sage.rings.complex_double import CDF
if PyInt_Check(py) or PyLong_Check(py):
return ZZ(py)
elif PyFloat_Check(py):
return RDF(py)
elif PyComplex_Check(py):
return CDF(py)
else:
raise TypeError, "Not a scalar"
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)
True
sage: is_Element(QQ^3)
False
"""
return IS_INSTANCE(x, Element)
cdef class Element(sage_object.SageObject):
"""
Generic element of a structure. All other types of elements
(RingElement, 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.
"""
def __init__(self, parent):
r"""
INPUT:
- ``parent`` - a SageObject
"""
self._parent = parent
def _set_parent(self, parent):
r"""
INPUT:
- ``parent`` - a SageObject
"""
self._parent = parent
cdef _set_parent_c(self, Parent parent):
self._parent = parent
def _make_new_with_parent_c(self, Parent parent):
self._parent = parent
return self
def __getattr__(self, str name):
"""
Let cat be the category of the parent of ``self``. This
method emulates ``self`` being an instance of both ``Element``
and ``cat.element_class``, in that order, for attribute
lookup.
NOTE:
Attributes beginning with two underscores but not ending with
an unnderscore are considered private and are thus exempted
from the lookup in ``cat.element_class``.
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 EuclideanDomains.element_class.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_with_category.element_class.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'
We test that "private" attributes are not requested from the element class
of the category (trac ticket #10467)::
sage: C = EuclideanDomains()
sage: P.<x> = QQ[]
sage: C.element_class.__foo = 'bar'
sage: x.parent() in C
True
sage: x.__foo
Traceback (most recent call last):
...
AttributeError: 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint' object has no attribute '__foo'
"""
if (name.startswith('__') and not name.endswith('_')):
raise AttributeError, AttributeErrorMessage(self, name)
cdef Parent P = self._parent or self.parent()
if P is None or P._category is None:
raise AttributeError, AttributeErrorMessage(self, name)
return getattr_from_other_class(self, P._category.element_class, name)
def __dir__(self):
"""
Let cat be the category of the parent of ``self``. This method
emulates ``self`` being an instance of both ``Element`` and
``cat.element_class``, in that order, for attribute directory.
EXAMPLES::
sage: dir(1/2)
['N', ..., 'is_idempotent', 'is_integral', ...]
Caveat: dir on Integer's and some other extension types seem to ignore __dir__::
sage: 1.__dir__()
['N', ..., 'is_idempotent', 'is_integral', ...]
sage: dir(1) # todo: not implemented
['N', ..., 'is_idempotent', 'is_integral', ...]
"""
from sage.structure.parent import dir_with_other_class
return dir_with_other_class(self, self.parent().category().element_class)
def _repr_(self):
return "Generic element of a structure"
def __getstate__(self):
"""
Returns 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__ring': Multivariate Polynomial Ring in x, y over Rational Field, '_Ideal_generic__gens': (x^2 - y^2 + 1,)})
"""
return (self._parent, self.__dict__)
def __setstate__(self, state):
"""
Initializes 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):
"""
Returns 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__
res = cls.__new__(cls)
res._set_parent(self._parent)
try:
res.__dict__ = self.__dict__.copy()
except AttributeError:
pass
return res
def __hash__(self):
return hash(str(self))
def _im_gens_(self, codomain, im_gens):
"""
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
cdef base_extend_c(self, Parent R):
if HAS_DICTIONARY(self):
return self.base_extend(R)
else:
return self.base_extend_c_impl(R)
cdef base_extend_c_impl(self, Parent R):
cdef Parent V
V = self._parent.base_extend(R)
return (<Parent>V)._coerce_c(self)
def base_extend(self, R):
return self.base_extend_c_impl(R)
def base_ring(self):
"""
Returns the base ring of this element's parent (if that makes sense).
"""
return self._parent.base_ring()
def category(self):
from sage.categories.all import Elements
return Elements(self._parent)
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
"""
from sage.categories.objects import Objects
tester = self._tester(**options)
sage_object.SageObject._test_category(self, tester = tester)
category = self.category()
if not is_extension_type(self.__class__):
tester.assert_(isinstance(self, self.parent().category().element_class))
else:
tester.assert_(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):
"""
Returns 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 hasattr(self,'__call__'):
return self
parent=self._parent
try:
ngens = parent.ngens()
except (AttributeError, NotImplementedError, TypeError):
return self
variables=[]
for i in xrange(0,ngens):
gen=parent.gen(i)
if kwds.has_key(str(gen)):
variables.append(kwds[str(gen)])
elif in_dict and in_dict.has_key(gen):
variables.append(in_dict[gen])
else:
variables.append(gen)
return self(*variables)
def numerical_approx (self, prec=None, digits=None):
"""
Return a numerical approximation of x with at least prec bits of
precision.
EXAMPLES::
sage: (2/3).n()
0.666666666666667
sage: pi.n(digits=10)
3.141592654
sage: pi.n(prec=20) # 20 bits
3.1416
"""
import sage.misc.functional
return sage.misc.functional.numerical_approx(self, prec=prec, digits=digits)
n=numerical_approx
N=n
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: 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)
mpf('4.14159265358979323846264338327933')
sage: (1+pi)._mpmath_(10)
mpf('4.140625')
sage: (1+pi)._mpmath_(mp.prec)
mpf('4.14159265358979323846264338327933')
"""
return self.n(prec)._mpmath_(prec=prec)
def substitute(self,in_dict=None,**kwds):
"""
This is an alias for self.subs().
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.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(in_dict,**kwds)
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()
False
"""
if self._parent.is_atomic_repr():
return True
s = str(self)
return s.find("+") == -1 and s.find("-") == -1 and s.find(" ") == -1
def __nonzero__(self):
r"""
Return True if self does not equal self.parent()(0).
Note that this is automatically called when converting to
boolean, as in the conditional of an if or while statement.
TESTS:
Verify that #5185 is fixed.
::
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
sage: (v+w).__nonzero__()
False
"""
return self != self._parent.zero_element()
def is_zero(self):
"""
Return True if self equals self.parent()(0). The default
implementation is to fall back to 'not self.__nonzero__'.
.. warning::
Do not re-implement this method in your subclass but
implement __nonzero__ instead.
"""
return not self
def _cmp_(left, right):
return left._cmp_c_impl(right)
cdef _cmp(left, right):
"""
Compare left and right.
"""
global coercion_model
cdef int r
if not have_same_parent(left, right):
if left is None or left is Ellipsis:
return -1
elif right is None or right is Ellipsis:
return 1
try:
_left, _right = coercion_model.canonical_coercion(left, right)
if PY_IS_NUMERIC(_left):
return cmp(_left, _right)
else:
return _left._cmp_(_right)
except TypeError:
r = cmp(type(left), type(right))
if r == 0:
r = -1
return r
else:
if HAS_DICTIONARY(left):
left_cmp = left._cmp_
if PY_TYPE_CHECK(left_cmp, MethodType):
return left_cmp(right)
return left._cmp_c_impl(right)
def _richcmp_(left, right, op):
return left._richcmp(right, op)
cdef _richcmp(left, right, int op):
"""
Compare left and right, according to the comparison operator op.
"""
global coercion_model
cdef int r
if not have_same_parent(left, right):
if left is None or left is Ellipsis:
return _rich_to_bool(op, -1)
elif right is None or right is Ellipsis:
return _rich_to_bool(op, 1)
try:
_left, _right = coercion_model.canonical_coercion(left, right)
if PY_IS_NUMERIC(_left):
return _rich_to_bool(op, cmp(_left, _right))
else:
return _left._richcmp_(_right, op)
except (TypeError, NotImplementedError):
r = cmp(type(left), type(right))
if r == 0:
r = -1
from sage.rings.integer import Integer
try:
if PY_TYPE_CHECK(left, Element) and isinstance(right, (int, float, Integer)) and not right:
right = (<Element>left)._parent(right)
elif PY_TYPE_CHECK(right, Element) and isinstance(left, (int, float, Integer)) and not left:
left = (<Element>right)._parent(left)
else:
return _rich_to_bool(op, r)
except TypeError:
return _rich_to_bool(op, r)
if HAS_DICTIONARY(left):
left_cmp = left.__cmp__
if PY_TYPE_CHECK(left_cmp, MethodType):
return _rich_to_bool(op, left_cmp(right))
return left._richcmp_c_impl(right, op)
cdef _rich_to_bool(self, int op, int r):
return _rich_to_bool(op, r)
def __richcmp__(left, right, int op):
return (<Element>left)._richcmp(right, op)
def __cmp__(left, right):
return (<Element>left)._cmp(right)
cdef _richcmp_c_impl(left, Element right, int op):
if (<Element>left)._richcmp_c_impl == Element._richcmp_c_impl and \
(<Element>left)._cmp_c_impl == Element._cmp_c_impl:
if op == Py_EQ:
return left is right
elif op == Py_NE:
return left is not right
return left._rich_to_bool(op, left._cmp_c_impl(right))
cdef int _cmp_c_impl(left, Element right) except -2:
raise NotImplementedError, "BUG: sort algorithm for elements of '%s' not implemented"%right.parent()
cdef inline bint _rich_to_bool(int op, int r):
if op == Py_LT:
return (r < 0)
elif op == Py_EQ:
return (r == 0)
elif op == Py_GT:
return (r > 0)
elif op == Py_LE:
return (r <= 0)
elif op == Py_NE:
return (r != 0)
elif op == Py_GE:
return (r >= 0)
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)
True
sage: is_ModuleElement((QQ^3).0)
True
sage: is_ModuleElement('a')
False
"""
return IS_INSTANCE(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 trac ticket #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, can not 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.
EXAMPLE:
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: cython_code = ["from sage.structure.element cimport Element, ElementWithCachedMethod",
... "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)",
... " def __cmp__(left, right):",
... " return (<Element>left)._cmp(right)",
... " def __richcmp__(left, right, op):",
... " return (<Element>left)._richcmp(right,op)",
... " cdef int _cmp_c_impl(left, Element right) except -2:",
... " return cmp(left.x,right.x)",
... " 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)",
... " def __cmp__(left, right):",
... " return (<Element>left)._cmp(right)",
... " def __richcmp__(left, right, op):",
... " return (<Element>left)._richcmp(right,op)",
... " cdef int _cmp_c_impl(left, Element right) except -2:",
... " return cmp(left.x,right.x)",
... " 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.all import cached_method, cached_in_parent_method, Category, 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: 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
True
sage: type(e) == type(ebroken)
False
sage: ebroken.element_via_parent_test() is e.element_via_parent_test()
True
However, the cache of the other inherited method breaks, although the method
as such works::
sage: ebroken.element_cache_test()
<-5>
sage: ebroken.element_cache_test() is ebroken.element_cache_test()
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()
sage: e.element_via_parent_test.clear_cache()
sage: b is ebroken.element_via_parent_test()
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)
sage: epython.element_cache_test()
<-5>
sage: epython.element_cache_test() is epython.element_cache_test()
True
"""
def __getattr__(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.
EXAMPLE::
sage: cython_code = ["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.all import cached_method, lazy_attribute, Category, 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 <%d>'%self.x"]
sage: cython('\n'.join(cython_code))
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
"""
if name.startswith('__') and not name.endswith('_'):
raise AttributeError, AttributeErrorMessage(self, name)
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.
"""
def __add__(left, right):
"""
Top-level addition operator for ModuleElements.
See extensive documentation at the top of element.pyx.
"""
if have_same_parent(left, right):
if (<RefPyObject *>left).ob_refcnt < inplace_threshold:
return (<ModuleElement>left)._iadd_(<ModuleElement>right)
else:
return (<ModuleElement>left)._add_(<ModuleElement>right)
global coercion_model
return coercion_model.bin_op(left, right, add)
cpdef ModuleElement _add_(left, ModuleElement right):
raise TypeError, arith_error_message(left, right, add)
def __iadd__(ModuleElement self, right):
if have_same_parent(self, right):
if (<RefPyObject *>self).ob_refcnt <= inplace_threshold:
return self._iadd_(<ModuleElement>right)
else:
return self._add_(<ModuleElement>right)
else:
global coercion_model
return coercion_model.bin_op(self, right, iadd)
cpdef ModuleElement _iadd_(self, ModuleElement right):
return self._add_(right)
def __sub__(left, right):
"""
Top-level subtraction operator for ModuleElements.
See extensive documentation at the top of element.pyx.
"""
if have_same_parent(left, right):
if (<RefPyObject *>left).ob_refcnt < inplace_threshold:
return (<ModuleElement>left)._isub_(<ModuleElement>right)
else:
return (<ModuleElement>left)._sub_(<ModuleElement>right)
global coercion_model
return coercion_model.bin_op(left, right, sub)
cpdef ModuleElement _sub_(left, ModuleElement right):
return left._add_(-right)
def __isub__(ModuleElement self, right):
if have_same_parent(self, right):
if (<RefPyObject *>self).ob_refcnt <= inplace_threshold:
return self._isub_(<ModuleElement>right)
else:
return self._sub_(<ModuleElement>right)
else:
global coercion_model
return coercion_model.bin_op(self, right, isub)
cpdef ModuleElement _isub_(self, ModuleElement right):
return self._sub_(right)
def __neg__(self):
"""
Top-level negation operator for ModuleElements, which
may choose to implement _neg_ rather than __neg__ for
consistency.
"""
return self._neg_()
cpdef ModuleElement _neg_(self):
global coercion_model
if self._parent._base is None:
return coercion_model.bin_op(-1, self, mul)
else:
return coercion_model.bin_op(self._parent._base(-1), self, mul)
def __mul__(left, right):
if PyInt_CheckExact(right):
return (<ModuleElement>left)._mul_long(PyInt_AS_LONG(right))
if PyInt_CheckExact(left):
return (<ModuleElement>right)._mul_long(PyInt_AS_LONG(left))
if have_same_parent(left, right):
raise arith_error_message(left, right, mul)
global coercion_model
return coercion_model.bin_op(left, right, mul)
def __imul__(left, right):
if have_same_parent(left, right):
raise TypeError
global coercion_model
return coercion_model.bin_op(left, right, imul)
cpdef ModuleElement _rmul_(self, RingElement left):
"""
Default module left scalar multiplication, which is to try to
canonically coerce the scalar to the integers and do that
multiplication, which is always defined.
Returning None indicates that this action is not implemented here.
"""
return None
cpdef ModuleElement _lmul_(self, RingElement right):
"""
Default module left scalar multiplication, which is to try to
canonically coerce the scalar to the integers and do that
multiplication, which is always defined.
Returning None indicates that this action is not implemented here.
"""
return None
cdef ModuleElement _mul_long(self, long n):
"""
Generic path for multiplying by a C long, assumed to commute.
"""
return coercion_model.bin_op(self, n, mul)
cpdef ModuleElement _ilmul_(self, RingElement right):
return self._lmul_(right)
cdef RingElement coerce_to_base_ring(self, x):
if PY_TYPE_CHECK(x, Element) and (<Element>x)._parent is self._parent._base:
return x
try:
return self._parent._base._coerce_c(x)
except AttributeError:
return self._parent._base(x)
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
def is_MonoidElement(x):
"""
Return True if x is of type MonoidElement.
"""
return IS_INSTANCE(x, MonoidElement)
cdef class MonoidElement(Element):
"""
Generic element of a monoid.
"""
def __mul__(left, right):
"""
Top-level multiplication operator for monoid elements.
See extensive documentation at the top of element.pyx.
"""
global coercion_model
if have_same_parent(left, right):
return (<MonoidElement>left)._mul_(<MonoidElement>right)
try:
return coercion_model.bin_op(left, right, mul)
except TypeError, msg:
if isinstance(left, (int, long)) and left==1:
return right
elif isinstance(right, (int, long)) and right==1:
return left
raise
cpdef MonoidElement _mul_(left, MonoidElement right):
"""
Cython classes should override this function to implement multiplication.
See extensive documentation at the top of element.pyx.
"""
raise TypeError
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
def __pow__(self, n, dummy):
"""
Return the (integral) power of self.
"""
if dummy is not None:
raise RuntimeError, "__pow__ dummy argument not used"
return generic_power_c(self,n,None)
def __nonzero__(self):
return True
def is_AdditiveGroupElement(x):
"""
Return True if x is of type AdditiveGroupElement.
"""
return IS_INSTANCE(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"
cpdef ModuleElement _rmul_(self, RingElement left):
return self._lmul_(left)
cpdef ModuleElement _lmul_(self, RingElement right):
"""
Default module left scalar multiplication, which is to try to
canonically coerce the scalar to the integers and do that
multiplication, which is always defined.
Returning None indicates this action is not implemented.
"""
return None
def is_MultiplicativeGroupElement(x):
"""
Return True if x is of type MultiplicativeGroupElement.
"""
return IS_INSTANCE(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()
def _add_(self, x):
raise ArithmeticError, "addition not defined in a multiplicative group"
def __div__(left, right):
if have_same_parent(left, right):
return left._div_(right)
global coercion_model
return coercion_model.bin_op(left, right, div)
cpdef MultiplicativeGroupElement _div_(self, MultiplicativeGroupElement right):
"""
Cython classes should override this function to implement division.
See extensive documentation at the top of element.pyx.
"""
return self * ~right
def __invert__(self):
if self.is_one():
return self
return 1/self
def is_RingElement(x):
"""
Return True if x is of type RingElement.
"""
return IS_INSTANCE(x, RingElement)
cdef class RingElement(ModuleElement):
def is_one(self):
return self == self._parent.one_element()
def __add__(left, right):
"""
Top-level addition operator for RingElements.
See extensive documentation at the top of element.pyx.
"""
if have_same_parent(left, right):
if (<RefPyObject *>left).ob_refcnt < inplace_threshold:
return (<ModuleElement>left)._iadd_(<ModuleElement>right)
else:
return (<ModuleElement>left)._add_(<ModuleElement>right)
if PyInt_CheckExact(right):
return (<RingElement>left)._add_long(PyInt_AS_LONG(right))
elif PyInt_CheckExact(left):
return (<RingElement>right)._add_long(PyInt_AS_LONG(left))
return coercion_model.bin_op(left, right, add)
cdef RingElement _add_long(self, long n):
"""
Generic path for adding a C long, assumed to commute.
"""
return coercion_model.bin_op(self, n, add)
def __sub__(left, right):
"""
Top-level subtraction operator for RingElements.
See extensive documentation at the top of element.pyx.
"""
cdef long n
if have_same_parent(left, right):
if (<RefPyObject *>left).ob_refcnt < inplace_threshold:
return (<ModuleElement>left)._isub_(<ModuleElement>right)
else:
return (<ModuleElement>left)._sub_(<ModuleElement>right)
if PyInt_CheckExact(right):
n = PyInt_AS_LONG(right)
if (n == 0) | (<unsigned long>n != 0 - <unsigned long>n):
return (<RingElement>left)._add_long(-n)
return coercion_model.bin_op(left, right, sub)
cpdef ModuleElement _lmul_(self, RingElement right):
return None
cpdef ModuleElement _rmul_(self, RingElement left):
return None
def __mul__(left, right):
"""
Top-level multiplication operator for ring elements.
See extensive documentation at the top of element.pyx.
AUTHOR:
- Gonzalo Tornaria (2007-06-25) - write base-extending test cases and fix them
TESTS:
Here we test (scalar * vector) multiplication::
sage: x, y = var('x, y')
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]))
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]))
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]))
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]))
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]))
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]))
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]))
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]))
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](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'
Here we test (scalar * matrix) multiplication::
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]))
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]))
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]))
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]))
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]))
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]))
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]))
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]))
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](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'
"""
if have_same_parent(left, right):
if (<RefPyObject *>left).ob_refcnt < inplace_threshold:
return (<RingElement>left)._imul_(<RingElement>right)
else:
return (<RingElement>left)._mul_(<RingElement>right)
if PyInt_CheckExact(right):
return (<ModuleElement>left)._mul_long(PyInt_AS_LONG(right))
elif PyInt_CheckExact(left):
return (<ModuleElement>right)._mul_long(PyInt_AS_LONG(left))
return coercion_model.bin_op(left, right, mul)
cpdef RingElement _mul_(self, RingElement right):
"""
Cython classes should override this function to implement multiplication.
See extensive documentation at the top of element.pyx.
"""
raise TypeError, arith_error_message(self, right, mul)
def __imul__(left, right):
if have_same_parent(left, right):
if (<RefPyObject *>left).ob_refcnt <= inplace_threshold:
return (<RingElement>left)._imul_(<RingElement>right)
else:
return (<RingElement>left)._mul_(<RingElement>right)
global coercion_model
return coercion_model.bin_op(left, right, imul)
cpdef RingElement _imul_(RingElement self, RingElement right):
return self._mul_(right)
def __pow__(self, n, dummy):
"""
Return the (integral) power of self.
EXAMPLE::
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):
...
RuntimeError: __pow__ dummy argument not used
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)
Traceback (most recent call last):
...
NotImplementedError: non-integral exponents not supported
TESTS::
These aren't testing this code, but they are probably good to have around.
sage: 2r**(SR(2)-1-1r)
1
sage: 2r^(1/2)
sqrt(2)
Exponent overflow should throw an OverflowError (trac #2956)::
sage: K.<x,y> = AA[]
sage: x^(2^64 + 12345)
Traceback (most recent call last):
...
OverflowError: Exponent overflow (2147483648).
Another example from trac #2956; this should overflow on x32
and succeed on x64::
sage: K.<x,y> = ZZ[]
sage: (x^12345)^54321
x^670592745 # 64-bit
Traceback (most recent call last): # 32-bit
... # 32-bit
OverflowError: Exponent overflow (670592745). # 32-bit
"""
if dummy is not None:
raise RuntimeError, "__pow__ dummy argument not used"
return generic_power_c(self,n,None)
def __truediv__(self, right):
if not PY_TYPE_CHECK(self, Element):
return coercion_model.bin_op(self, right, div)
return self.__div__(right)
def __div__(self, right):
"""
Top-level multiplication operator for ring elements.
See extensive documentation at the top of element.pyx.
"""
if have_same_parent(self, right):
if (<RefPyObject *>self).ob_refcnt < inplace_threshold:
return (<RingElement>self)._idiv_(<RingElement>right)
else:
return (<RingElement>self)._div_(<RingElement>right)
global coercion_model
return coercion_model.bin_op(self, right, div)
cpdef RingElement _div_(self, RingElement right):
"""
Cython classes should override this function to implement division.
See extensive documentation at the top of element.pyx.
"""
try:
return self._parent.fraction_field()(self, right)
except AttributeError:
if not right:
raise ZeroDivisionError, "Cannot divide by zero"
else:
raise TypeError, arith_error_message(self, right, div)
def __idiv__(self, right):
"""
Top-level division operator for ring elements.
See extensive documentation at the top of element.pyx.
"""
if have_same_parent(self, right):
if (<RefPyObject *>self).ob_refcnt <= inplace_threshold:
return (<RingElement>self)._idiv_(<RingElement>right)
else:
return (<RingElement>self)._div_(<RingElement>right)
global coercion_model
return coercion_model.bin_op(self, right, idiv)
cpdef RingElement _idiv_(RingElement self, RingElement right):
return self._div_(right)
def __invert__(self):
if self.is_one():
return self
return 1/self
def order(self):
"""
Return the additive order of self.
This is deprecated; use ``additive_order`` instead.
EXAMPLES::
sage: a = Integers(12)(5)
sage: a.order()
doctest... DeprecationWarning: The function order is deprecated for ring elements; use additive_order or multiplicative_order instead.
12
"""
from sage.misc.misc import deprecation
deprecation("The function order is deprecated for ring elements; use additive_order or multiplicative_order instead.")
return self.additive_order()
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, or raise
``ArithmeticError`` otherwise.
"""
if not self.is_unit():
raise ArithmeticError, "self (=%s) must be a unit to have a multiplicative order."
raise NotImplementedError
def is_unit(self):
if self == 1 or self == -1:
return True
raise NotImplementedError
def is_nilpotent(self):
"""
Return True if self is nilpotent, i.e., some power of self
is 0.
"""
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()
1.00000000000000
sage: ZZ(-1).abs()
1
sage: CC(I).abs()
1.00000000000000
sage: Mod(-15, 37).abs()
Traceback (most recent call last):
...
ArithmeticError: absolute valued not defined on integers modulo n.
"""
return self.__abs__()
def is_CommutativeRingElement(x):
"""
Return True if x is of type CommutativeRingElement.
"""
return IS_INSTANCE(x, CommutativeRingElement)
cdef class CommutativeRingElement(RingElement):
def _im_gens_(self, codomain, im_gens):
if len(im_gens) == 1 and self._parent.gen(0) == 1:
return codomain(self)
raise NotImplementedError
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.
"""
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
Ticket \#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))
Traceback (most recent call last):
...
ZeroDivisionError: reduction modulo right not defined.
If x has different parent than `self`, they are first coerced to a
common parent if possible. If this coercion fails, it returns a
TypeError. This fixes \#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
return (x % self) == 0
else:
global coercion_model
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.)
EXAMPLE: 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
EXAMPLE: Univariate polynomials
::
sage: R.<x> = PolynomialRing(QQ)
sage: f = x^3 + x + 1
sage: f.mod(x + 1)
-1
When little is implemented about a given ring, then mod may
return simply return `f`. For example, reduction is not
implemented for `\ZZ[x]` yet. (TODO!)
sage: R.<x> = PolynomialRing(ZZ)
sage: f = x^3 + x + 1
sage: f.mod(x + 1)
x^3 + x + 1
EXAMPLE: 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)
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) )
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) )
9*z^4
In this example `y` is eliminated::
sage: (x^2 + y^2 + z^2).mod( (x^3, y - z) )
x^2 + 2*z^2
"""
from sage.rings.all import is_Ideal
if not is_Ideal(I) or not I.ring() is self._parent:
I = self._parent.ideal(I)
return I.reduce(self)
def is_square(self, root=False):
"""
Returns whether or not ring element is a square. If the optional
argument root is True, then also returns the square root (or None,
if the it is not a square).
INPUT:
- ``root`` - whether or not to also return a square
root (default: False)
OUTPUT:
- ``bool`` - whether or not a square
- ``object`` - (optional) an actual square root if
found, and None otherwise.
EXAMPLES::
sage: R.<x> = PolynomialRing(QQ)
sage: f = 12*(x+1)^2 * (x+3)^2
sage: f.is_square()
False
sage: f.is_square(root=True)
(False, None)
sage: h = f/3
sage: h.is_square()
True
sage: h.is_square(root=True)
(True, 2*x^2 + 8*x + 6)
.. NOTE:
This is the is_square implementation for general commutative ring
elements. It's implementation is to raise a NotImplementedError.
The function definition is here to show what functionality is expected and
provide a general framework.
"""
raise NotImplementedError("is_square() not implemented for elements of %s" %self.parent())
def sqrt(self, extend = True, all = False, name=None ):
"""
It computes the square root.
INPUT:
- ``extend`` - Whether to make a ring extension containing a square root if self is not a square (default: True)
- ``all`` - Whether to return a list of all square roots or just a square root (default: False)
- ``name`` - Required when extend=True and self is not a square. This will be the name of the generator extension.
OUTPUT:
- if all=False it returns a square root. (throws an error if extend=False and self is not a square)
- if all=True it returns a list of all the square roots (could be empty if extend=False and self is not a square)
ALGORITHM:
It uses is_square(root=true) for the hard part of the work, the rest is just wrapper code.
EXAMPLES::
sage: R.<x> = ZZ[]
sage: (x^2).sqrt()
x
sage: f=x^2-4*x+4; f.sqrt(all=True)
[x - 2, -x + 2]
sage: sqrtx=x.sqrt(name="y"); sqrtx
y
sage: sqrtx^2
x
sage: x.sqrt(all=true,name="y")
[y, -y]
sage: x.sqrt(extend=False,all=True)
[]
sage: x.sqrt()
Traceback (most recent call last):
...
TypeError: Polynomial is not a square. You must specify the name of the square root when using the default extend = True
sage: x.sqrt(extend=False)
Traceback (most recent call last):
...
ValueError: trying to take square root of non-square x with extend = False
TESTS::
sage: f = (x+3)^2; f.sqrt()
x + 3
sage: f = (x+3)^2; f.sqrt(all=True)
[x + 3, -x - 3]
sage: f = (x^2 - x + 3)^2; f.sqrt()
x^2 - x + 3
sage: f = (x^2 - x + 3)^6; f.sqrt()
x^6 - 3*x^5 + 12*x^4 - 19*x^3 + 36*x^2 - 27*x + 27
sage: g = (R.random_element(15))^2
sage: g.sqrt()^2 == g
True
sage: R.<x> = GF(250037)[]
sage: f = x^2/(x+1)^2; f.sqrt()
x/(x + 1)
sage: f = 9 * x^4 / (x+1)^2; f.sqrt()
3*x^2/(x + 1)
sage: f = 9 * x^4 / (x+1)^2; f.sqrt(all=True)
[3*x^2/(x + 1), 250034*x^2/(x + 1)]
sage: R.<x> = QQ[]
sage: a = 2*(x+1)^2 / (2*(x-1)^2); a.sqrt()
(2*x + 2)/(2*x - 2)
sage: sqrtx=(1/x).sqrt(name="y"); sqrtx
y
sage: sqrtx^2
1/x
sage: (1/x).sqrt(all=true,name="y")
[y, -y]
sage: (1/x).sqrt(extend=False,all=True)
[]
sage: (1/(x^2-1)).sqrt()
Traceback (most recent call last):
...
TypeError: Polynomial is not a square. You must specify the name of the square root when using the default extend = True
sage: (1/(x^2-3)).sqrt(extend=False)
Traceback (most recent call last):
...
ValueError: trying to take square root of non-square 1/(x^2 - 3) with extend = False
"""
from sage.rings.integral_domain import is_IntegralDomain
P=self._parent
is_sqr, sq_rt = self.is_square( root = True )
if is_sqr:
if all:
if not is_IntegralDomain(P):
raise NotImplementedError('sqrt() with all=True is only implemented for integral domains, not for %s' % P)
if P.characteristic()==2 or sq_rt==0:
return [ sq_rt ]
return [ sq_rt, -sq_rt ]
return sq_rt
if not is_IntegralDomain(P):
raise NotImplementedError('sqrt() of non squares is only implemented for integral domains, not for %s' % P)
if not extend:
if all:
return []
raise ValueError, 'trying to take square root of non-square %s with extend = False' % self
if name == None:
raise TypeError ("Polynomial is not a square. You must specify the name of the square root when using the default extend = True")
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
PY = PolynomialRing(P,'y')
y = PY.gen()
sq_rt = PY.quotient(y**2-self, names = name)(y)
if all:
if P.characteristic() == 2:
return [ sq_rt ]
return [ sq_rt, -sq_rt ]
return sq_rt
cdef class Vector(ModuleElement):
cdef bint is_sparse_c(self):
raise NotImplementedError
cdef bint is_dense_c(self):
raise NotImplementedError
def __imul__(left, right):
if have_same_parent(left, right):
return (<Vector>left)._dot_product_(<Vector>right)
global coercion_model
return coercion_model.bin_op(left, right, imul)
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: x, y = var('x, y')
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]))
Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(ZZ[x],[1,2,3,4])*vector(QQ,[1,2,3,4]))
Univariate Polynomial Ring in x over Rational Field
sage: parent(vector(QQ,[1,2,3,4])*vector(ZZ[x][y],[1,2,3,4]))
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]))
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]))
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]))
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]))
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]))
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
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: 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]))
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]))
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]))
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]))
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]))
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]))
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]))
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]))
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],[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: 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))
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))
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))
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))
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))
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))
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))
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))
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],[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'
"""
if have_same_parent(left, right):
return (<Vector>left)._dot_product_(<Vector>right)
global coercion_model
return coercion_model.bin_op(left, right, mul)
cpdef Element _dot_product_(Vector left, Vector right):
raise TypeError, arith_error_message(left, right, mul)
cpdef Vector _pairwise_product_(Vector left, Vector right):
raise TypeError, "unsupported operation for '%s' and '%s'"%(parent_c(left), parent_c(right))
def __div__(self, right):
if PY_IS_NUMERIC(right):
right = py_scalar_to_element(right)
if PY_TYPE_CHECK(right, RingElement):
return self.__mul__(~right)
if PY_TYPE_CHECK(right, Vector):
try:
W = 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"
raise TypeError, arith_error_message(self, right, div)
def _magma_init_(self, magma):
"""
Return string that evaluates in Magma to something equivalent
to this vector.
EXAMPLES::
sage: v = vector([1,2,3])
sage: v._magma_init_(magma) # optional - magma
'_sage_[...]![1,2,3]'
sage: mv = magma(v); mv # optional - magma
(1 2 3)
sage: mv.Type() # optional - magma
ModTupRngElt
sage: mv.Parent() # optional - magma
Full RSpace of degree 3 over Integer Ring
sage: v = vector(QQ, [1/2, 3/4, 5/6])
sage: mv = magma(v); mv # optional - magma
(1/2 3/4 5/6)
sage: mv.Type() # optional - magma
ModTupFldElt
sage: mv.Parent() # optional - magma
Full Vector space of degree 3 over Rational Field
A more demanding example::
sage: R.<x,y,z> = QQ[]
sage: v = vector([x^3, y, 2/3*z + x/y])
sage: magma(v) # optional - magma
( x^3 y (2/3*y*z + x)/y)
sage: magma(v).Parent() # optional - magma
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):
return IS_INSTANCE(x, Vector)
cdef class Matrix(AlgebraElement):
cdef bint is_sparse_c(self):
raise NotImplementedError
cdef bint is_dense_c(self):
raise NotImplementedError
def __imul__(left, right):
if have_same_parent(left, right):
return (<Matrix>left)._matrix_times_matrix_(<Matrix>right)
else:
global coercion_model
return coercion_model.bin_op(left, right, imul)
cpdef RingElement _mul_(left, RingElement right):
"""
TESTS::
sage: m = matrix
sage: a = m([[m([[1,2],[3,4]]),m([[5,6],[7,8]])],[m([[9,10],[11,12]]),m([[13,14],[15,16]])]])
sage: 3*a
[[ 3 6]
[ 9 12] [15 18]
[21 24]]
[[27 30]
[33 36] [39 42]
[45 48]]
sage: m = matrix
sage: a = m([[m([[1,2],[3,4]]),m([[5,6],[7,8]])],[m([[9,10],[11,12]]),m([[13,14],[15,16]])]])
sage: a*3
[[ 3 6]
[ 9 12] [15 18]
[21 24]]
[[27 30]
[33 36] [39 42]
[45 48]]
"""
if have_same_parent(left, right):
return (<Matrix>left)._matrix_times_matrix_(<Matrix>right)
else:
global coercion_model
return coercion_model.bin_op(left, right, mul)
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: x, y = var('x, y')
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]))
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]))
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]))
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]))
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]))
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]))
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]))
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]))
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],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'
Here we test (matrix * vector) multiplication::
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]))
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]))
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]))
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]))
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]))
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]))
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]))
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]))
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],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: 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))
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))
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))
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))
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))
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))
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))
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))
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],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'
"""
if have_same_parent(left, right):
return (<Matrix>left)._matrix_times_matrix_(<Matrix>right)
else:
global coercion_model
return coercion_model.bin_op(left, right, mul)
cdef Vector _vector_times_matrix_(matrix_right, Vector vector_left):
raise TypeError
cdef Vector _matrix_times_vector_(matrix_left, Vector vector_right):
raise TypeError
cdef Matrix _matrix_times_matrix_(left, Matrix right):
raise TypeError
def is_Matrix(x):
return IS_INSTANCE(x, Matrix)
def is_IntegralDomainElement(x):
"""
Return True if x is of type IntegralDomainElement.
"""
return IS_INSTANCE(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.
"""
return IS_INSTANCE(x, DedekindDomainElement)
cdef class DedekindDomainElement(IntegralDomainElement):
pass
def is_PrincipalIdealDomainElement(x):
"""
Return True if x is of type PrincipalIdealDomainElement.
"""
return IS_INSTANCE(x, PrincipalIdealDomainElement)
cdef class PrincipalIdealDomainElement(DedekindDomainElement):
def lcm(self, right):
"""
Returns the least common multiple of self and right.
"""
if not PY_TYPE_CHECK(right, Element) or not ((<Element>right)._parent is self._parent):
return coercion_model.bin_op(self, right, lcm)
return self._lcm(right)
def gcd(self, right):
"""
Returns the gcd of self and right, or 0 if both are 0.
"""
if not PY_TYPE_CHECK(right, Element) or not ((<Element>right)._parent is self._parent):
return coercion_model.bin_op(self, right, gcd)
return self._gcd(right)
def xgcd(self, right):
r"""
Return the extended gcd of self and other, i.e., elements `r, s, t` such that
.. math::
r = s \cdot self + t \cdot other.
.. note::
There is no guarantee on minimality of the cofactors. In
the integer case, see documentation for Integer._xgcd() to
obtain minimal cofactors.
"""
if not PY_TYPE_CHECK(right, Element) or not ((<Element>right)._parent is self._parent):
return coercion_model.bin_op(self, right, xgcd)
return self._xgcd(right)
PY_SET_TP_NEW(EuclideanDomainElement, Element)
def is_EuclideanDomainElement(x):
"""
Return True if x is of type EuclideanDomainElement.
"""
return IS_INSTANCE(x, EuclideanDomainElement)
cdef class EuclideanDomainElement(PrincipalIdealDomainElement):
def degree(self):
raise NotImplementedError
def _gcd(self, other):
"""
Return the greatest common divisor of self and other.
Algorithm 3.2.1 in Cohen, GTM 138.
"""
A = self
B = other
while not B.is_zero():
Q, R = A.quo_rem(B)
A = B
B = R
return A
def leading_coefficient(self):
raise NotImplementedError
def quo_rem(self, other):
raise NotImplementedError
def __divmod__(self, other):
"""
Return the quotient and remainder of self divided by other.
EXAMPLES::
sage: divmod(5,3)
(1, 2)
sage: divmod(25r,12)
(2, 1)
sage: divmod(25,12r)
(2, 1)
"""
if PY_TYPE_CHECK(self, Element):
return self.quo_rem(other)
else:
x, y = canonical_coercion(self, other)
return x.quo_rem(y)
def __floordiv__(self,right):
"""
Quotient of division of self by other. This is denoted //.
"""
Q, _ = self.quo_rem(right)
return Q
def __mod__(self, other):
"""
Remainder of division of self by other.
EXAMPLES::
sage: R.<x> = ZZ[]
sage: x % (x+1)
-1
sage: (x**3 + x - 1) % (x**2 - 1)
2*x - 1
"""
_, R = self.quo_rem(other)
return R
def is_FieldElement(x):
"""
Return True if x is of type FieldElement.
"""
return IS_INSTANCE(x, FieldElement)
cdef class FieldElement(CommutativeRingElement):
def __floordiv__(self, other):
return self / other
def is_unit(self):
"""
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 _gcd(self, FieldElement other):
"""
Return the greatest common divisor of self and other.
"""
if self.is_zero() and other.is_zero():
return self
else:
return self._parent(1)
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 _xgcd(self, FieldElement other):
R = self._parent
if not self.is_zero():
return R(1), ~self, R(0)
elif not other.is_zero():
return R(1), R(0), ~self
else:
return self, self, self
def quo_rem(self, right):
r"""
Return the quotient and remainder obtained by dividing `self` by
`other`. Since this element lives in a field, the remainder is always
zero and the quotient is `self/right`.
TESTS:
Test if #8671 is fixed::
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_element()
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 non-zero values.
EXAMPLES::
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 is_AlgebraElement(x):
"""
Return True if x is of type AlgebraElement.
"""
return IS_INSTANCE(x, AlgebraElement)
cdef class AlgebraElement(RingElement):
pass
def is_CommutativeAlgebraElement(x):
"""
Return True if x is of type CommutativeAlgebraElement.
"""
return IS_INSTANCE(x, CommutativeAlgebraElement)
cdef class CommutativeAlgebraElement(CommutativeRingElement):
pass
def is_InfinityElement(x):
"""
Return True if x is of type InfinityElement.
"""
return IS_INSTANCE(x, InfinityElement)
cdef class InfinityElement(RingElement):
def __invert__(self):
from sage.rings.all import ZZ
return ZZ(0)
cdef class PlusInfinityElement(InfinityElement):
pass
cdef class MinusInfinityElement(InfinityElement):
pass
include "coerce.pxi"
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]])
sage: canonical_coercion(A,1)
(
[0 1] [1 0]
[1 0], [0 1]
)
"""
global coercion_model
return coercion_model.canonical_coercion(x,y)
cpdef bin_op(x, y, op):
global coercion_model
return coercion_model.bin_op(x,y,op)
def coerce(Parent p, x):
try:
return p._coerce_c(x)
except AttributeError:
return p(x)
def coerce_cmp(x,y):
global coercion_model
cdef int c
try:
x, y = coercion_model.canonical_coercion(x, y)
return cmp(x,y)
except TypeError:
c = cmp(type(x), type(y))
if c == 0: c = -1
return c
cdef class CoercionModel:
"""
Most basic coercion scheme. If it doesn't already match, throw an error.
"""
cpdef canonical_coercion(self, x, y):
if parent_c(x) is parent_c(y):
return x,y
raise TypeError, "no common canonical parent for objects with parents: '%s' and '%s'"%(parent_c(x), parent_c(y))
cpdef bin_op(self, x, y, op):
if parent_c(x) is parent_c(y):
return op(x,y)
raise TypeError, arith_error_message(x,y,op)
import coerce
cdef CoercionModel coercion_model = coerce.CoercionModel_cache_maps()
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_cache_maps object at ...>
"""
return coercion_model
def set_coercion_model(cm):
global coercion_model
coercion_model = cm
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()
cdef class NamedBinopMethod:
"""
A decorator to be used on binary operation methods that should operate
on elements of the same parent. If the parents of the arguments differ,
coercion is performed, then the method is re-looked up by name on the
first argument.
In short, using the ``NamedBinopMethod`` (alias ``coerce_binop``) decorator
on a method gives it the exact same semantics of the basic arithmetic
operations like ``_add_``, ``_sub_``, etc. in that both operands are
guaranteed to have exactly the same parent.
"""
cdef _self
cdef _func
cdef _name
def __init__(self, func, name=None, obj=None):
"""
TESTS::
sage: from sage.structure.element import NamedBinopMethod
sage: NamedBinopMethod(gcd)(12, 15)
3
"""
self._func = func
if name is None:
if isinstance(func, types.FunctionType):
name = func.func_name
if isinstance(func, types.UnboundMethodType):
name = func.im_func.func_name
else:
name = func.__name__
self._name = name
self._self = obj
def __call__(self, x, y=None, **kwds):
"""
TESTS::
sage: from sage.structure.element import NamedBinopMethod
sage: test_func = NamedBinopMethod(lambda x, y, **kwds: (x, y, kwds), '_add_')
Calls func directly if the two arguments have the same parent::
sage: test_func(1, 2)
(1, 2, {})
Passes through coercion and does a method lookup if the
left operand is not the same::
sage: test_func(0.5, 1)
(0.500000000000000, 1.00000000000000, {})
sage: test_func(1, 2, algorithm='fast')
(1, 2, {'algorithm': 'fast'})
sage: test_func(1, 1/2)
3/2
"""
if y is None:
if self._self is None:
self._func(x, **kwds)
else:
x, y = self._self, x
if not have_same_parent(x, y):
old_x = x
x,y = coercion_model.canonical_coercion(x, y)
if old_x is x:
return self._func(x,y, *kwds)
else:
return getattr(x, self._name)(y, **kwds)
else:
return self._func(x,y, **kwds)
def __get__(self, obj, objtype):
"""
Used to transform from an unbound to a bound method.
TESTS::
sage: from sage.structure.element import NamedBinopMethod
sage: R.<x> = ZZ[]
sage: isinstance(x.quo_rem, NamedBinopMethod)
True
sage: x.quo_rem(x)
(1, 0)
sage: type(x).quo_rem(x,x)
(1, 0)
"""
return NamedBinopMethod(self._func, self._name, obj)
def _sage_doc_(self):
"""
Return the docstring of the wrapped object for introspection.
EXAMPLES::
sage: from sage.structure.element import NamedBinopMethod
sage: g = NamedBinopMethod(gcd)
sage: from sage.misc.sageinspect import sage_getdoc
sage: sage_getdoc(g) == sage_getdoc(gcd)
True
"""
return sageinspect._sage_getdoc_unformatted(self._func)
def _sage_src_(self):
"""
Returns the source of the wrapped object for introspection.
EXAMPLES::
sage: from sage.structure.element import NamedBinopMethod
sage: g = NamedBinopMethod(gcd)
sage: 'def gcd(' in g._sage_src_()
True
"""
return sageinspect.sage_getsource(self._func)
def _sage_argspec_(self):
"""
Returns the argspec of the wrapped object for introspection.
EXAMPLES::
sage: from sage.structure.element import NamedBinopMethod
sage: g = NamedBinopMethod(gcd)
sage: g._sage_argspec_()
ArgSpec(args=['a', 'b'], varargs=None, keywords='kwargs', defaults=(None,))
"""
return sageinspect.sage_getargspec(self._func)
coerce_binop = NamedBinopMethod
def lcm(x,y):
from sage.rings.arith import lcm
return lcm(x,y)
def gcd(x,y):
from sage.rings.arith import gcd
return gcd(x,y)
def xgcd(x,y):
from sage.rings.arith import xgcd
return xgcd(x,y)
def generic_power(a, n, one=None):
"""
Computes `a^n`, where `n` is an integer, and `a` is an object which
supports multiplication. Optionally an additional argument,
which is used in the case that n == 0:
- ``one`` - the "unit" element, returned directly (can be anything)
If this is not supplied, int(1) is returned.
EXAMPLES::
sage: from sage.structure.element import generic_power
sage: generic_power(int(12),int(0))
1
sage: generic_power(int(0),int(100))
0
sage: generic_power(Integer(10),Integer(0))
1
sage: generic_power(Integer(0),Integer(23))
0
sage: sum([generic_power(2,i) for i in range(17)]) #test all 4-bit combinations
131071
sage: F = Zmod(5)
sage: a = generic_power(F(2), 5); a
2
sage: a.parent() is F
True
sage: a = generic_power(F(1), 2)
sage: a.parent() is F
True
sage: generic_power(int(5), 0)
1
"""
return generic_power_c(a,n,one)
cdef generic_power_c(a, nn, one):
try:
n = PyNumber_Index(nn)
except TypeError:
try:
from sage.rings.integer import Integer
n = int(Integer(nn))
except TypeError:
raise NotImplementedError, "non-integral exponents not supported"
if not n:
if one is None:
if PY_TYPE_CHECK(a, Element):
return (<Element>a)._parent.one()
try:
try:
return a.parent().one()
except AttributeError:
return type(a)(1)
except:
return 1
else:
return one
elif n < 0:
a = ~a
n = -n
if n < 4:
if n == 1:
return a
elif n == 2:
return a*a
elif n == 3:
return a*a*a
aa = a*a
if aa == a:
return a
m = n & 1
n = n >> 1
apow = aa
while n&1 == 0:
apow = apow*apow
n = n >> 1
power = apow
n = n >> 1
if m:
power = power * a
while n != 0:
apow = apow*apow
if n&1 != 0:
power = power*apow
n = n >> 1
return power
