def __init__(self, prec=53):
self._prec = int(prec)
from sage.categories.fields import Fields
ParentWithGens.__init__(self,
self._real_field(), ('I', ),
False,
category=Fields())
def __init__(self, coordinate_patch = None):
"""
Construct the algebra of differential forms on a given coordinate patch.
See ``DifferentialForms`` for details.
INPUT::
- ``coordinate_patch`` -- Coordinate patch where the algebra lives.
If no coordinate patch is given, a default coordinate patch with
coordinates (x, y, z) is used.
EXAMPLES::
sage: p, q = var('p, q')
sage: U = CoordinatePatch((p, q)); U
Open subset of R^2 with coordinates p, q
sage: F = DifferentialForms(U); F
Algebra of differential forms in the variables p, q
"""
from sage.categories.graded_algebras_with_basis \
import GradedAlgebrasWithBasis
from sage.structure.parent_gens import ParentWithGens
if not coordinate_patch:
x, y, z = var('x, y, z')
coordinate_patch = CoordinatePatch((x, y, z))
if not isinstance(coordinate_patch, CoordinatePatch):
raise TypeError("%s not a valid Coordinate Patch" % coordinate_patch)
self._patch = coordinate_patch
ParentWithGens.__init__(self, SR, \
category = GradedAlgebrasWithBasis(SR))
def __init__(self, prec=53):
"""
Initialize ``self``.
TESTS::
sage: C = ComplexField(200)
sage: C.category()
Join of Category of fields and Category of infinite sets and Category of complete metric spaces
sage: TestSuite(C).run()
sage: CC.is_field()
True
sage: CC.is_finite()
False
"""
self._prec = int(prec)
from sage.categories.fields import Fields
ParentWithGens.__init__(
self,
self._real_field(), ('I', ),
False,
category=Fields().Infinite().Metric().Complete())
# self._populate_coercion_lists_()
self._populate_coercion_lists_(
coerce_list=[RRtoCC(self._real_field(), self)])
def __init__(self, group, base_ring = IntegerRing()):
r""" Create the given group algebra.
INPUT:
-- (Group) group: a generic group.
-- (Ring) base_ring: a commutative ring.
OUTPUT:
-- a GroupAlgebra instance.
EXAMPLES::
sage: from sage.algebras.group_algebra import GroupAlgebra
doctest:1: DeprecationWarning:...
sage: GroupAlgebra(GL(3, GF(7)))
Group algebra of group "General Linear Group of degree 3 over Finite
Field of size 7" over base ring Integer Ring
sage: GroupAlgebra(1)
Traceback (most recent call last):
...
TypeError: "1" is not a group
sage: GroupAlgebra(SU(2, GF(4, 'a')), IntegerModRing(12)).category()
Category of group algebras over Ring of integers modulo 12
"""
if not base_ring.is_commutative():
raise NotImplementedError("Base ring must be commutative")
if not is_Group(group):
raise TypeError('"%s" is not a group' % group)
ParentWithGens.__init__(self, base_ring, category = GroupAlgebras(base_ring))
self._formal_sum_module = FormalSums(base_ring)
self._group = group
def __init__(self):
"""
TEST::
sage: sage.rings.infinity.InfinityRing_class() is sage.rings.infinity.InfinityRing_class() is InfinityRing
True
"""
ParentWithGens.__init__(self, self, names=('oo', ), normalize=False)
def __init__(self):
"""
TEST::
sage: sage.rings.infinity.InfinityRing_class() is sage.rings.infinity.InfinityRing_class() is InfinityRing
True
"""
ParentWithGens.__init__(self, self, names=('oo',), normalize=False)
def __init__(self):
"""
TESTS::
sage: sage.rings.infinity.UnsignedInfinityRing_class() is sage.rings.infinity.UnsignedInfinityRing_class() is UnsignedInfinityRing
True
"""
ParentWithGens.__init__(self, self, names=("oo",), normalize=False)
def __init__(self):
"""
Initialize ``self``.
TESTS::
sage: sage.rings.infinity.UnsignedInfinityRing_class() is sage.rings.infinity.UnsignedInfinityRing_class() is UnsignedInfinityRing
True
"""
ParentWithGens.__init__(self, self, names=('oo', ), normalize=False)
def __init__(self, prec=53):
"""
Initialize ``self``.
EXAMPLES::
sage: ComplexIntervalField()
Complex Interval Field with 53 bits of precision
sage: ComplexIntervalField(200)
Complex Interval Field with 200 bits of precision
"""
self._prec = int(prec)
from sage.categories.fields import Fields
ParentWithGens.__init__(self, self._real_field(), ('I',), False, category = Fields())
def __init__(self, prec=53):
"""
Initialize ``self``.
EXAMPLES::
sage: ComplexIntervalField()
Complex Interval Field with 53 bits of precision
sage: ComplexIntervalField(200)
Complex Interval Field with 200 bits of precision
"""
self._prec = int(prec)
from sage.categories.fields import Fields
ParentWithGens.__init__(self, self._real_field(), ('I',), False, category = Fields())
def __init__(self, polynomial, names, category=CAT):
"""
Create a function field defined as an extension of another
function field by adjoining a root of a univariate polynomial.
INPUT:
- ``polynomial`` -- a univariate polynomial over a function field
- ``names`` -- variable names (as a tuple of length 1 or string)
- ``category`` -- a category (defaults to category of function fields)
EXAMPLES::
We create an extension of function fields::
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L = K.extension(y^5 - x^3 - 3*x + x*y); L
Function field in y defined by y^5 + x*y - x^3 - 3*x
Note the type::
sage: type(L)
<class 'sage.rings.function_field.function_field.FunctionField_polymod_with_category'>
We can set the variable name, which doesn't have to be y::
sage: L.<w> = K.extension(y^5 - x^3 - 3*x + x*y); L
Function field in w defined by y^5 + x*y - x^3 - 3*x
"""
from sage.rings.polynomial.all import is_Polynomial
if names is None:
names = (polynomial.variable_name(), )
if not is_Polynomial(polynomial):
raise TypeError, "polynomial must be a polynomial"
if polynomial.degree() <= 0:
raise ValueError, "polynomial must have positive degree"
base_field = polynomial.base_ring()
if not isinstance(base_field, FunctionField):
raise TypeError, "polynomial must be over a function"
self._base_field = base_field
self._polynomial = polynomial
ParentWithGens.__init__(self,
base_field,
names=names,
category=category)
self._hash = hash(polynomial)
self._ring = self._polynomial.parent()
self._populate_coercion_lists_(coerce_list=[base_field, self._ring])
self._gen = self(self._ring.gen())
def __init__(self, prec=53):
"""
TESTS::
sage: C = ComplexField(200)
sage: C.category()
Category of fields
sage: TestSuite(C).run()
"""
self._prec = int(prec)
from sage.categories.fields import Fields
ParentWithGens.__init__(self, self._real_field(), ('I',), False, category = Fields())
# self._populate_coercion_lists_()
self._populate_coercion_lists_(coerce_list=[complex_number.RRtoCC(self._real_field(), self)])
def __init__(self, prec=53):
"""
TESTS::
sage: C = ComplexField(200)
sage: C.category()
Category of fields
sage: TestSuite(C).run()
"""
self._prec = int(prec)
from sage.categories.fields import Fields
ParentWithGens.__init__(self, self._real_field(), ('I',), False, category = Fields())
# self._populate_coercion_lists_()
self._populate_coercion_lists_(coerce_list=[complex_number.RRtoCC(self._real_field(), self)])
def __init__(self, polynomial, names, category=CAT):
"""
Create a function field defined as an extension of another
function field by adjoining a root of a univariate polynomial.
INPUT:
- ``polynomial`` -- a univariate polynomial over a function field
- ``names`` -- variable names (as a tuple of length 1 or string)
- ``category`` -- a category (defaults to category of function fields)
EXAMPLES::
We create an extension of function fields::
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L = K.extension(y^5 - x^3 - 3*x + x*y); L
Function field in y defined by y^5 + x*y - x^3 - 3*x
Note the type::
sage: type(L)
<class 'sage.rings.function_field.function_field.FunctionField_polymod_with_category'>
We can set the variable name, which doesn't have to be y::
sage: L.<w> = K.extension(y^5 - x^3 - 3*x + x*y); L
Function field in w defined by y^5 + x*y - x^3 - 3*x
"""
from sage.rings.polynomial.all import is_Polynomial
if names is None:
names = (polynomial.variable_name(), )
if not is_Polynomial(polynomial):
raise TypeError, "polynomial must be a polynomial"
if polynomial.degree() <= 0:
raise ValueError, "polynomial must have positive degree"
base_field = polynomial.base_ring()
if not isinstance(base_field, FunctionField):
raise TypeError, "polynomial must be over a function"
self._base_field = base_field
self._polynomial = polynomial
ParentWithGens.__init__(self, base_field,
names=names, category = category)
self._hash = hash(polynomial)
self._ring = self._polynomial.parent()
self._populate_coercion_lists_(coerce_list=[base_field, self._ring])
self._gen = self(self._ring.gen())
def __init__(self):
"""
Initialize ``self``.
TESTS::
sage: sage.rings.infinity.UnsignedInfinityRing_class() is sage.rings.infinity.UnsignedInfinityRing_class() is UnsignedInfinityRing
True
Sage can understand SymPy's complex infinity (:trac:`17493`)::
sage: import sympy
sage: SR(sympy.zoo)
Infinity
"""
ParentWithGens.__init__(self, self, names=('oo', ), normalize=False)
def __init__(self, prec=53):
"""
Initialize ``self``.
TESTS::
sage: C = ComplexField(200)
sage: C.category()
Join of Category of fields and Category of complete metric spaces
sage: TestSuite(C).run()
"""
self._prec = int(prec)
from sage.categories.fields import Fields
ParentWithGens.__init__(self, self._real_field(), ('I',), False, category=Fields().Metric().Complete())
# self._populate_coercion_lists_()
self._populate_coercion_lists_(coerce_list=[RRtoCC(self._real_field(), self)])
def __init__(self):
"""
Initialize ``self``.
TESTS::
sage: sage.rings.infinity.UnsignedInfinityRing_class() is sage.rings.infinity.UnsignedInfinityRing_class() is UnsignedInfinityRing
True
Sage can understand SymPy's complex infinity (:trac:`17493`)::
sage: import sympy
sage: SR(sympy.zoo)
Infinity
"""
ParentWithGens.__init__(self, self, names=('oo',), normalize=False)
def __init__(self, coordinate_patch=None):
"""
Construct the algebra of differential forms on a given coordinate patch.
See ``DifferentialForms`` for details.
INPUT:
- ``coordinate_patch`` -- Coordinate patch where the algebra lives.
If no coordinate patch is given, a default coordinate patch with
coordinates (x, y, z) is used.
EXAMPLES::
sage: p, q = var('p, q')
sage: U = CoordinatePatch((p, q)); U
doctest:...: DeprecationWarning: Use Manifold instead.
See http://trac.sagemath.org/24444 for details.
Open subset of R^2 with coordinates p, q
sage: F = DifferentialForms(U); F
doctest:...: DeprecationWarning: For the set of differential forms of
degree p, use U.diff_form_module(p), where U is the base manifold
(type U.diff_form_module? for details).
See http://trac.sagemath.org/24444 for details.
Algebra of differential forms in the variables p, q
"""
from sage.categories.graded_algebras_with_basis \
import GradedAlgebrasWithBasis
from sage.structure.parent_gens import ParentWithGens
from sage.misc.superseded import deprecation
deprecation(
24444, 'For the set of differential forms of degree p, ' +
'use U.diff_form_module(p), where U is the base ' +
'manifold (type U.diff_form_module? for details).')
if not coordinate_patch:
x, y, z = var('x, y, z')
coordinate_patch = CoordinatePatch((x, y, z))
if not isinstance(coordinate_patch, CoordinatePatch):
raise TypeError("%s not a valid Coordinate Patch" %
coordinate_patch)
self._patch = coordinate_patch
ParentWithGens.__init__(self, SR, \
category = GradedAlgebrasWithBasis(SR))
def __init__(self, coordinate_patch = None):
"""
Construct the algebra of differential forms on a given coordinate patch.
See ``DifferentialForms`` for details.
INPUT:
- ``coordinate_patch`` -- Coordinate patch where the algebra lives.
If no coordinate patch is given, a default coordinate patch with
coordinates (x, y, z) is used.
EXAMPLES::
sage: p, q = var('p, q')
sage: U = CoordinatePatch((p, q)); U
doctest:...: DeprecationWarning: Use Manifold instead.
See http://trac.sagemath.org/24444 for details.
Open subset of R^2 with coordinates p, q
sage: F = DifferentialForms(U); F
doctest:...: DeprecationWarning: For the set of differential forms of
degree p, use U.diff_form_module(p), where U is the base manifold
(type U.diff_form_module? for details).
See http://trac.sagemath.org/24444 for details.
Algebra of differential forms in the variables p, q
"""
from sage.categories.graded_algebras_with_basis \
import GradedAlgebrasWithBasis
from sage.structure.parent_gens import ParentWithGens
from sage.misc.superseded import deprecation
deprecation(24444, 'For the set of differential forms of degree p, ' +
'use U.diff_form_module(p), where U is the base ' +
'manifold (type U.diff_form_module? for details).')
if not coordinate_patch:
x, y, z = var('x, y, z')
coordinate_patch = CoordinatePatch((x, y, z))
if not isinstance(coordinate_patch, CoordinatePatch):
raise TypeError("%s not a valid Coordinate Patch" % coordinate_patch)
self._patch = coordinate_patch
ParentWithGens.__init__(self, SR, \
category = GradedAlgebrasWithBasis(SR))
def __init__(self, constant_field, names, category=CAT):
"""
Create a rational function field in one variable.
INPUT:
- ``constant_field`` -- an arbitrary field
- ``names`` -- a string or tuple of length 1
- ``category`` -- default: FunctionFields()
EXAMPLES::
sage: K.<t> = FunctionField(CC); K
Rational function field in t over Complex Field with 53 bits of precision
sage: K.category()
Category of function fields
sage: FunctionField(QQ[I], 'alpha')
Rational function field in alpha over Number Field in I with defining polynomial x^2 + 1
Must be over a field::
sage: FunctionField(ZZ, 't')
Traceback (most recent call last):
...
TypeError: constant_field must be a field
"""
if names is None:
raise ValueError, "variable name must be specified"
elif not isinstance(names, tuple):
names = (names, )
if not constant_field.is_field():
raise TypeError, "constant_field must be a field"
ParentWithGens.__init__(self, self, names=names, category = category)
R = constant_field[names[0]]
self._hash = hash((constant_field, names))
self._constant_field = constant_field
self._ring = R
self._field = R.fraction_field()
self._populate_coercion_lists_(coerce_list=[self._field])
self._gen = self(R.gen())
def __init__(self, constant_field, names, category=CAT):
"""
Create a rational function field in one variable.
INPUT:
- ``constant_field`` -- an arbitrary field
- ``names`` -- a string or tuple of length 1
- ``category`` -- default: FunctionFields()
EXAMPLES::
sage: K.<t> = FunctionField(CC); K
Rational function field in t over Complex Field with 53 bits of precision
sage: K.category()
Category of function fields
sage: FunctionField(QQ[I], 'alpha')
Rational function field in alpha over Number Field in I with defining polynomial x^2 + 1
Must be over a field::
sage: FunctionField(ZZ, 't')
Traceback (most recent call last):
...
TypeError: constant_field must be a field
"""
if names is None:
raise ValueError, "variable name must be specified"
elif not isinstance(names, tuple):
names = (names, )
if not constant_field.is_field():
raise TypeError, "constant_field must be a field"
ParentWithGens.__init__(self, self, names=names, category=category)
R = constant_field[names[0]]
self._hash = hash((constant_field, names))
self._constant_field = constant_field
self._ring = R
self._field = R.fraction_field()
self._populate_coercion_lists_(coerce_list=[self._field])
self._gen = self(R.gen())
def __init__(self):
r"""
We create the rational numbers `\QQ`, and call a few functions::
sage: Q = RationalField(); Q
Rational Field
sage: Q.characteristic()
0
sage: Q.is_field()
True
sage: Q.category()
Join of Category of number fields
and Category of quotient fields
and Category of metric spaces
sage: Q.zeta()
-1
We next illustrate arithmetic in `\QQ`.
::
sage: Q('49/7')
7
sage: type(Q('49/7'))
<type 'sage.rings.rational.Rational'>
sage: a = Q('19/374'); a
19/374
sage: b = Q('17/371'); b
17/371
sage: a + b
13407/138754
sage: b + a
13407/138754
sage: a * b
19/8162
sage: b * a
19/8162
sage: a - b
691/138754
sage: b - a
-691/138754
sage: a / b
7049/6358
sage: b / a
6358/7049
sage: b < a
True
sage: a < b
False
Next finally illustrate arithmetic with automatic coercion. The
types that coerce into the rational field include ``str, int,
long, Integer``.
::
sage: a + Q('17/371')
13407/138754
sage: a * 374
19
sage: 374 * a
19
sage: a/19
1/374
sage: a + 1
393/374
TESTS::
sage: TestSuite(QQ).run()
sage: QQ.variable_name()
'x'
sage: QQ.variable_names()
('x',)
sage: QQ._element_constructor_((2, 3))
2/3
sage: QQ.is_finite()
False
sage: QQ.is_field()
True
"""
from sage.categories.basic import QuotientFields
from sage.categories.number_fields import NumberFields
ParentWithGens.__init__(
self, self, category=[QuotientFields().Metric(),
NumberFields()])
self._assign_names(('x', ), normalize=False) # ?????
self._populate_coercion_lists_(init_no_parent=True)
def __init__(self, prec=53):
self._prec = int(prec)
from sage.categories.fields import Fields
ParentWithGens.__init__(self, self._real_field(), ('I',), False, category = Fields())
def __init__(self, base_ring, num_gens, name_list,
order='negdeglex', default_prec=10, sparse=False):
"""
Initializes a multivariate power series ring. See PowerSeriesRing
for complete documentation.
INPUT
- ``base_ring`` - a commutative ring
- ``num_gens`` - number of generators
- ``name_list`` - List of indeterminate names or a single name.
If a single name is given, indeterminates will be this name
followed by a number from 0 to num_gens - 1. If a list is
given, these will be the indeterminate names and the length
of the list must be equal to num_gens.
- ``order`` - ordering of variables; default is
negative degree lexicographic
- ``default_prec`` - The default total-degree precision for
elements. The default value of default_prec is 10.
- ``sparse`` - whether or not power series are sparse
EXAMPLES::
sage: R.<t,u,v> = PowerSeriesRing(QQ)
sage: g = 1 + v + 3*u*t^2 - 2*v^2*t^2
sage: g = g.add_bigoh(5); g
1 + v + 3*t^2*u - 2*t^2*v^2 + O(t, u, v)^5
sage: g in R
True
"""
order = TermOrder(order,num_gens)
self._term_order = order
if not base_ring.is_commutative():
raise TypeError("Base ring must be a commutative ring.")
n = int(num_gens)
if n < 0:
raise ValueError("Multivariate Polynomial Rings must have more than 0 variables.")
self._ngens = n
self._has_singular = False #cannot convert to Singular by default
ParentWithGens.__init__(self, base_ring, name_list)
Nonexact.__init__(self, default_prec)
# underlying polynomial ring in which to represent elements
self._poly_ring_ = PolynomialRing(base_ring, self.variable_names(), sparse=sparse, order=order)
# because sometimes PowerSeriesRing_generic calls self.__poly_ring
self._PowerSeriesRing_generic__poly_ring = self._poly_ring()
# background univariate power series ring
self._bg_power_series_ring = PowerSeriesRing(self._poly_ring_, 'Tbg', sparse=sparse, default_prec=default_prec)
self._bg_indeterminate = self._bg_power_series_ring.gen()
## use the following in PowerSeriesRing_generic.__call__
self._PowerSeriesRing_generic__power_series_class = MPowerSeries
self._is_sparse = sparse
self._params = (base_ring, num_gens, name_list,
order, default_prec, sparse)
self._populate_coercion_lists_()
def __init__(self):
r"""
We create the rational numbers `\QQ`, and call a few functions::
sage: Q = RationalField(); Q
Rational Field
sage: Q.characteristic()
0
sage: Q.is_field()
True
sage: Q.category()
Category of quotient fields
sage: Q.zeta()
-1
We next illustrate arithmetic in `\QQ`.
::
sage: Q('49/7')
7
sage: type(Q('49/7'))
<type 'sage.rings.rational.Rational'>
sage: a = Q('19/374'); b = Q('17/371'); print a, b
19/374 17/371
sage: a + b
13407/138754
sage: b + a
13407/138754
sage: a * b
19/8162
sage: b * a
19/8162
sage: a - b
691/138754
sage: b - a
-691/138754
sage: a / b
7049/6358
sage: b / a
6358/7049
sage: b < a
True
sage: a < b
False
Next finally illustrate arithmetic with automatic coercion. The
types that coerce into the rational field include ``str, int,
long, Integer``.
::
sage: a + Q('17/371')
13407/138754
sage: a * 374
19
sage: 374 * a
19
sage: a/19
1/374
sage: a + 1
393/374
TESTS::
sage: TestSuite(QQ).run()
sage: QQ.variable_name()
'x'
sage: QQ.variable_names()
('x',)
"""
from sage.categories.basic import QuotientFields
ParentWithGens.__init__(self, self, category = QuotientFields())
self._assign_names(('x',),normalize=False) # ???
self._populate_coercion_lists_(element_constructor=rational.Rational, init_no_parent=True)
def __init__(self,
base_ring,
num_gens,
name_list,
order='negdeglex',
default_prec=10,
sparse=False):
"""
Initializes a multivariate power series ring. See PowerSeriesRing
for complete documentation.
INPUT
- ``base_ring`` - a commutative ring
- ``num_gens`` - number of generators
- ``name_list`` - List of indeterminate names or a single name.
If a single name is given, indeterminates will be this name
followed by a number from 0 to num_gens - 1. If a list is
given, these will be the indeterminate names and the length
of the list must be equal to num_gens.
- ``order`` - ordering of variables; default is
negative degree lexicographic
- ``default_prec`` - The default total-degree precision for
elements. The default value of default_prec is 10.
- ``sparse`` - whether or not power series are sparse
EXAMPLES::
sage: R.<t,u,v> = PowerSeriesRing(QQ)
sage: g = 1 + v + 3*u*t^2 - 2*v^2*t^2
sage: g = g.add_bigoh(5); g
1 + v + 3*t^2*u - 2*t^2*v^2 + O(t, u, v)^5
sage: g in R
True
"""
order = TermOrder(order, num_gens)
self._term_order = order
if not base_ring.is_commutative():
raise TypeError("Base ring must be a commutative ring.")
n = int(num_gens)
if n < 0:
raise ValueError(
"Multivariate Polynomial Rings must have more than 0 variables."
)
self._ngens = n
self._has_singular = False #cannot convert to Singular by default
ParentWithGens.__init__(self, base_ring, name_list)
Nonexact.__init__(self, default_prec)
# underlying polynomial ring in which to represent elements
self._poly_ring_ = PolynomialRing(base_ring,
self.variable_names(),
sparse=sparse,
order=order)
# because sometimes PowerSeriesRing_generic calls self.__poly_ring
self._PowerSeriesRing_generic__poly_ring = self._poly_ring()
# background univariate power series ring
self._bg_power_series_ring = PowerSeriesRing(self._poly_ring_,
'Tbg',
sparse=sparse,
default_prec=default_prec)
self._bg_indeterminate = self._bg_power_series_ring.gen()
## use the following in PowerSeriesRing_generic.__call__
self._PowerSeriesRing_generic__power_series_class = MPowerSeries
self._is_sparse = sparse
self._params = (base_ring, num_gens, name_list, order, default_prec,
sparse)
self._populate_coercion_lists_()