def _coerce_map_from_(self, P):
r"""
Return whether ``P`` coerces into this symbolic subring.
INPUT:
- ``P`` -- a parent.
OUTPUT:
A boolean or ``None``.
TESTS::
sage: from sage.symbolic.subring import GenericSymbolicSubring
sage: GenericSymbolicSubring(vars=tuple()).has_coerce_map_from(SR) # indirect doctest # not tested see #19231
False
::
sage: from sage.symbolic.subring import SymbolicSubring
sage: C = SymbolicSubring(no_variables=True)
sage: C.has_coerce_map_from(ZZ) # indirect doctest
True
sage: C.has_coerce_map_from(QQ) # indirect doctest
True
sage: C.has_coerce_map_from(RR) # indirect doctest
True
sage: C.has_coerce_map_from(RIF) # indirect doctest
True
sage: C.has_coerce_map_from(CC) # indirect doctest
True
sage: C.has_coerce_map_from(CIF) # indirect doctest
True
sage: C.has_coerce_map_from(AA) # indirect doctest
True
sage: C.has_coerce_map_from(QQbar) # indirect doctest
True
sage: C.has_coerce_map_from(SR) # indirect doctest
False
"""
if P == SR:
# Workaround; can be deleted once #19231 is fixed
return False
from sage.rings.real_mpfr import mpfr_prec_min
from sage.rings.all import (ComplexField, RLF, CLF, AA, QQbar,
InfinityRing)
from sage.rings.real_mpfi import is_RealIntervalField
from sage.rings.complex_interval_field import is_ComplexIntervalField
if isinstance(P, type):
return SR._coerce_map_from_(P)
elif RLF.has_coerce_map_from(P) or \
CLF.has_coerce_map_from(P) or \
AA.has_coerce_map_from(P) or \
QQbar.has_coerce_map_from(P):
return True
elif (P is InfinityRing or is_RealIntervalField(P)
or is_ComplexIntervalField(P)):
return True
elif ComplexField(mpfr_prec_min()).has_coerce_map_from(P):
return P not in (RLF, CLF, AA, QQbar)
def __getitem__(self, arg):
"""
Extend this ring by one or several elements to create a polynomial
ring, a power series ring, or an algebraic extension.
This is a convenience method intended primarily for interactive
use.
.. SEEALSO::
:func:`~sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing`,
:func:`~sage.rings.power_series_ring.PowerSeriesRing`,
:meth:`~sage.rings.ring.Ring.extension`,
:meth:`sage.rings.integer_ring.IntegerRing_class.__getitem__`,
:meth:`sage.rings.matrix_space.MatrixSpace.__getitem__`,
:meth:`sage.structure.parent.Parent.__getitem__`
EXAMPLES:
We create several polynomial rings::
sage: ZZ['x']
Univariate Polynomial Ring in x over Integer Ring
sage: QQ['x']
Univariate Polynomial Ring in x over Rational Field
sage: GF(17)['abc']
Univariate Polynomial Ring in abc over Finite Field of size 17
sage: GF(17)['a,b,c']
Multivariate Polynomial Ring in a, b, c over Finite Field of size 17
sage: GF(17)['a']['b']
Univariate Polynomial Ring in b over Univariate Polynomial Ring in a over Finite Field of size 17
We can create skew polynomial rings::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: k['x',Frob]
Skew Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
We can also create power series rings by using double brackets::
sage: QQ[['t']]
Power Series Ring in t over Rational Field
sage: ZZ[['W']]
Power Series Ring in W over Integer Ring
sage: ZZ[['x,y,z']]
Multivariate Power Series Ring in x, y, z over Integer Ring
sage: ZZ[['x','T']]
Multivariate Power Series Ring in x, T over Integer Ring
Use :func:`~sage.rings.fraction_field.Frac` or
:meth:`~sage.rings.ring.CommutativeRing.fraction_field` to obtain
the fields of rational functions and Laurent series::
sage: Frac(QQ['t'])
Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: Frac(QQ[['t']])
Laurent Series Ring in t over Rational Field
sage: QQ[['t']].fraction_field()
Laurent Series Ring in t over Rational Field
Note that the same syntax can be used to create number fields::
sage: QQ[I]
Number Field in I with defining polynomial x^2 + 1
sage: QQ[I].coerce_embedding()
Generic morphism:
From: Number Field in I with defining polynomial x^2 + 1
To: Complex Lazy Field
Defn: I -> 1*I
::
sage: QQ[sqrt(2)]
Number Field in sqrt2 with defining polynomial x^2 - 2
sage: QQ[sqrt(2)].coerce_embedding()
Generic morphism:
From: Number Field in sqrt2 with defining polynomial x^2 - 2
To: Real Lazy Field
Defn: sqrt2 -> 1.414213562373095?
::
sage: QQ[sqrt(2),sqrt(3)]
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
and orders in number fields::
sage: ZZ[I]
Order in Number Field in I with defining polynomial x^2 + 1
sage: ZZ[sqrt(5)]
Order in Number Field in sqrt5 with defining polynomial x^2 - 5
sage: ZZ[sqrt(2)+sqrt(3)]
Order in Number Field in a with defining polynomial x^4 - 10*x^2 + 1
Embeddings are found for simple extensions (when that makes sense)::
sage: QQi.<i> = QuadraticField(-1, 'i')
sage: QQ[i].coerce_embedding()
Generic morphism:
From: Number Field in i with defining polynomial x^2 + 1
To: Complex Lazy Field
Defn: i -> 1*I
TESTS:
A few corner cases::
sage: QQ[()]
Multivariate Polynomial Ring in no variables over Rational Field
sage: QQ[[]]
Traceback (most recent call last):
...
TypeError: power series rings must have at least one variable
These kind of expressions do not work::
sage: QQ['a,b','c']
Traceback (most recent call last):
...
ValueError: variable name 'a,b' is not alphanumeric
sage: QQ[['a,b','c']]
Traceback (most recent call last):
...
ValueError: variable name 'a,b' is not alphanumeric
sage: QQ[[['x']]]
Traceback (most recent call last):
...
TypeError: expected R[...] or R[[...]], not R[[[...]]]
Extension towers are built as follows and use distinct generator names::
sage: K = QQ[2^(1/3), 2^(1/2), 3^(1/3)]
sage: K
Number Field in a with defining polynomial x^3 - 2 over its base field
sage: K.base_field()
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
sage: K.base_field().base_field()
Number Field in b with defining polynomial x^3 - 3
Embeddings::
sage: QQ[I](I.pyobject())
I
sage: a = 10^100; expr = (2*a + sqrt(2))/(2*a^2-1)
sage: QQ[expr].coerce_embedding() is None
False
sage: QQ[sqrt(5)].gen() > 0
True
sage: expr = sqrt(2) + I*(cos(pi/4, hold=True) - sqrt(2)/2)
sage: QQ[expr].coerce_embedding()
Generic morphism:
From: Number Field in a with defining polynomial x^2 - 2
To: Real Lazy Field
Defn: a -> 1.414213562373095?
"""
def normalize_arg(arg):
if isinstance(arg, (tuple, list)):
# Allowing arbitrary iterables would create confusion, but we
# may want to support a few more.
return tuple(arg)
elif isinstance(arg, str):
return tuple(arg.split(','))
else:
return (arg,)
# 1. If arg is a list, try to return a power series ring.
if isinstance(arg, list):
if arg == []:
raise TypeError("power series rings must have at least one variable")
elif len(arg) == 1:
# R[["a,b"]], R[[(a,b)]]...
if isinstance(arg[0], list):
raise TypeError("expected R[...] or R[[...]], not R[[[...]]]")
elts = normalize_arg(arg[0])
else:
elts = normalize_arg(arg)
from sage.rings.power_series_ring import PowerSeriesRing
return PowerSeriesRing(self, elts)
if isinstance(arg, tuple):
from sage.categories.morphism import Morphism
if len(arg) == 2 and isinstance(arg[1], Morphism):
from sage.rings.polynomial.skew_polynomial_ring_constructor import SkewPolynomialRing
return SkewPolynomialRing(self, arg[1], names=arg[0])
# 2. Otherwise, if all specified elements are algebraic, try to
# return an algebraic extension
elts = normalize_arg(arg)
try:
minpolys = [a.minpoly() for a in elts]
except (AttributeError, NotImplementedError, ValueError, TypeError):
minpolys = None
if minpolys:
# how to pass in names?
names = tuple(_gen_names(elts))
if len(elts) == 1:
from sage.rings.all import CIF, CLF, RIF, RLF
elt = elts[0]
try:
iv = CIF(elt)
except (TypeError, ValueError):
emb = None
else:
# First try creating an ANRoot manually, because
# extension(..., embedding=CLF(expr)) (or
# ...QQbar(expr)) would normalize the expression in
# QQbar, which currently is VERY slow in many cases.
# This may fail when minpoly has close roots or elt is
# a complicated symbolic expression.
# TODO: Rewrite using #19362 and/or #17886 and/or
# #15600 once those issues are solved.
from sage.rings.qqbar import AlgebraicNumber, ANRoot
try:
elt = AlgebraicNumber(ANRoot(minpolys[0], iv))
except ValueError:
pass
# Force a real embedding when possible, to get the
# right ordered ring structure.
if (iv.imag().is_zero() or iv.imag().contains_zero()
and elt.imag().is_zero()):
emb = RLF(elt)
else:
emb = CLF(elt)
return self.extension(minpolys[0], names[0], embedding=emb)
try:
# Doing the extension all at once is best, if possible...
return self.extension(minpolys, names)
except (TypeError, ValueError):
# ...but we can also construct it iteratively
return reduce(lambda R, ext: R.extension(*ext), zip(minpolys, names), self)
# 2. Otherwise, try to return a polynomial ring
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
return PolynomialRing(self, elts)
def _coerce_map_from_(self, P):
r"""
Return whether ``P`` coerces into this symbolic subring.
INPUT:
- ``P`` -- a parent.
OUTPUT:
A boolean or ``None``.
TESTS::
sage: from sage.symbolic.subring import GenericSymbolicSubring
sage: GenericSymbolicSubring(vars=tuple()).has_coerce_map_from(SR) # indirect doctest # not tested see #19231
False
::
sage: from sage.symbolic.subring import SymbolicSubring
sage: C = SymbolicSubring(no_variables=True)
sage: C.has_coerce_map_from(ZZ) # indirect doctest
True
sage: C.has_coerce_map_from(QQ) # indirect doctest
True
sage: C.has_coerce_map_from(RR) # indirect doctest
True
sage: C.has_coerce_map_from(RIF) # indirect doctest
True
sage: C.has_coerce_map_from(CC) # indirect doctest
True
sage: C.has_coerce_map_from(CIF) # indirect doctest
True
sage: C.has_coerce_map_from(AA) # indirect doctest
True
sage: C.has_coerce_map_from(QQbar) # indirect doctest
True
sage: C.has_coerce_map_from(SR) # indirect doctest
False
"""
if P == SR:
# Workaround; can be deleted once #19231 is fixed
return False
from sage.rings.real_mpfr import mpfr_prec_min
from sage.rings.all import (ComplexField,
RLF, CLF, AA, QQbar, InfinityRing)
from sage.rings.real_mpfi import is_RealIntervalField
from sage.rings.complex_interval_field import is_ComplexIntervalField
if isinstance(P, type):
return SR._coerce_map_from_(P)
elif RLF.has_coerce_map_from(P) or \
CLF.has_coerce_map_from(P) or \
AA.has_coerce_map_from(P) or \
QQbar.has_coerce_map_from(P):
return True
elif (P is InfinityRing or
is_RealIntervalField(P) or is_ComplexIntervalField(P)):
return True
elif ComplexField(mpfr_prec_min()).has_coerce_map_from(P):
return P not in (RLF, CLF, AA, QQbar)
def __init__(self, point, dop=None):
"""
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.path import Point
sage: Dops, x, Dx = DifferentialOperators()
sage: [Point(z, Dx)
....: for z in [1, 1/2, 1+I, QQbar(I), RIF(1/3), CIF(1/3), pi,
....: RDF(1), CDF(I), 0.5r, 0.5jr, 10r, QQbar(1), AA(1/3)]]
[1, 1/2, I + 1, I, [0.333333333333333...], [0.333333333333333...],
3.141592653589794?, 1.000000000000000, 1.000000000000000*I,
0.5000000000000000, 0.5000000000000000*I, 10, 1, 1/3]
sage: Point(sqrt(2), Dx).iv()
[1.414...]
"""
SageObject.__init__(self)
from sage.rings.complex_double import ComplexDoubleField_class
from sage.rings.complex_field import ComplexField_class
from sage.rings.complex_interval_field import ComplexIntervalField_class
from sage.rings.real_double import RealDoubleField_class
from sage.rings.real_mpfi import RealIntervalField_class
from sage.rings.real_mpfr import RealField_class
point = sage.structure.coerce.py_scalar_to_element(point)
try:
parent = point.parent()
except AttributeError:
raise TypeError("unexpected value for point: " + repr(point))
if isinstance(point, Point):
self.value = point.value
elif isinstance(
parent,
(number_field_base.NumberField, RealBallField, ComplexBallField)):
self.value = point
elif QQ.has_coerce_map_from(parent):
self.value = QQ.coerce(point)
# must come before QQbar, due to a bogus coerce map (#14485)
elif parent is sage.symbolic.ring.SR:
try:
return self.__init__(point.pyobject(), dop)
except TypeError:
pass
try:
return self.__init__(QQbar(point), dop)
except (TypeError, ValueError, NotImplementedError):
pass
try:
self.value = RLF(point)
except (TypeError, ValueError):
self.value = CLF(point)
elif QQbar.has_coerce_map_from(parent):
alg = QQbar.coerce(point)
NF, val, hom = alg.as_number_field_element()
if NF is QQ:
self.value = QQ.coerce(val) # parent may be ZZ
else:
embNF = number_field.NumberField(NF.polynomial(),
NF.variable_name(),
embedding=hom(NF.gen()))
self.value = val.polynomial()(embNF.gen())
elif isinstance(
parent,
(RealField_class, RealDoubleField_class, RealIntervalField_class)):
self.value = RealBallField(point.prec())(point)
elif isinstance(parent, (ComplexField_class, ComplexDoubleField_class,
ComplexIntervalField_class)):
self.value = ComplexBallField(point.prec())(point)
else:
try:
self.value = RLF.coerce(point)
except TypeError:
self.value = CLF.coerce(point)
parent = self.value.parent()
assert (isinstance(
parent,
(number_field_base.NumberField, RealBallField, ComplexBallField))
or parent is RLF or parent is CLF)
self.dop = dop or point.dop
self.keep_value = False
def __init__(self, point, dop=None, singular=None, **kwds):
"""
INPUT:
- ``singular``: can be set to True to force this point to be considered
a singular point, even if this cannot be checked (e.g. because we only
have an enclosure)
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.path import Point
sage: Dops, x, Dx = DifferentialOperators()
sage: [Point(z, Dx)
....: for z in [1, 1/2, 1+I, QQbar(I), RIF(1/3), CIF(1/3), pi,
....: RDF(1), CDF(I), 0.5r, 0.5jr, 10r, QQbar(1), AA(1/3)]]
[1, 1/2, I + 1, I, [0.333333333333333...], [0.333333333333333...],
3.141592653589794?, ~1.0000, ~1.0000*I, ~0.50000, ~0.50000*I, 10,
1, 1/3]
sage: Point(sqrt(2), Dx).iv()
[1.414...]
sage: Point(RBF(0), (x-1)*x*Dx, singular=True).dist_to_sing()
1.000000000000000
"""
SageObject.__init__(self)
from sage.rings.complex_double import ComplexDoubleField_class
from sage.rings.complex_field import ComplexField_class
from sage.rings.complex_interval_field import ComplexIntervalField_class
from sage.rings.real_double import RealDoubleField_class
from sage.rings.real_mpfi import RealIntervalField_class
from sage.rings.real_mpfr import RealField_class
point = sage.structure.coerce.py_scalar_to_element(point)
try:
parent = point.parent()
except AttributeError:
raise TypeError("unexpected value for point: " + repr(point))
if isinstance(point, Point):
self.value = point.value
elif isinstance(parent, (RealBallField, ComplexBallField)):
self.value = point
elif isinstance(parent, number_field_base.NumberField):
_, hom = good_number_field(point.parent())
self.value = hom(point)
elif QQ.has_coerce_map_from(parent):
self.value = QQ.coerce(point)
elif QQbar.has_coerce_map_from(parent):
alg = QQbar.coerce(point)
NF, val, hom = alg.as_number_field_element()
if NF is QQ:
self.value = QQ.coerce(val) # parent may be ZZ
else:
embNF = number_field.NumberField(NF.polynomial(),
NF.variable_name(),
embedding=hom(NF.gen()))
self.value = val.polynomial()(embNF.gen())
elif isinstance(
parent,
(RealField_class, RealDoubleField_class, RealIntervalField_class)):
self.value = RealBallField(point.prec())(point)
elif isinstance(parent, (ComplexField_class, ComplexDoubleField_class,
ComplexIntervalField_class)):
self.value = ComplexBallField(point.prec())(point)
elif parent is sage.symbolic.ring.SR:
try:
return self.__init__(point.pyobject(), dop)
except TypeError:
pass
try:
return self.__init__(QQbar(point), dop)
except (TypeError, ValueError, NotImplementedError):
pass
try:
self.value = RLF(point)
except (TypeError, ValueError):
self.value = CLF(point)
else:
try:
self.value = RLF.coerce(point)
except TypeError:
self.value = CLF.coerce(point)
parent = self.value.parent()
assert (isinstance(
parent,
(number_field_base.NumberField, RealBallField, ComplexBallField))
or parent is RLF or parent is CLF)
if dop is None: # TBI
if isinstance(point, Point):
self.dop = point.dop
else:
self.dop = DifferentialOperator(dop.numerator())
self._force_singular = bool(singular)
self.options = kwds