本文整理汇总了Python中sage.rings.real_mpfr.is_RealField函数的典型用法代码示例。如果您正苦于以下问题：Python is_RealField函数的具体用法？Python is_RealField怎么用？Python is_RealField使用的例子？那么恭喜您, 这里精选的函数代码示例或许可以为您提供帮助。
在下文中一共展示了is_RealField函数的13个代码示例，这些例子默认根据受欢迎程度排序。您可以为喜欢或者感觉有用的代码点赞，您的评价将有助于系统推荐出更棒的Python代码示例。
示例1: can_convert_to_singular
def can_convert_to_singular(R):
"""
Returns True if this ring's base field or ring can be
represented in Singular, and the polynomial ring has at
least one generator. If this is True then this polynomial
ring can be represented in Singular.
The following base rings are supported: finite fields, rationals, number
fields, and real and complex fields.
EXAMPLES::
sage: from sage.rings.polynomial.polynomial_singular_interface import can_convert_to_singular
sage: can_convert_to_singular(PolynomialRing(QQ, names=['x']))
True
sage: can_convert_to_singular(PolynomialRing(QQ, names=[]))
False
"""
if R.ngens() == 0:
return False;
base_ring = R.base_ring()
return ( sage.rings.finite_rings.constructor.is_FiniteField(base_ring)
or is_RationalField(base_ring)
or (base_ring.is_prime_field() and base_ring.characteristic() <= 2147483647)
or is_RealField(base_ring)
or is_ComplexField(base_ring)
or is_RealDoubleField(base_ring)
or is_ComplexDoubleField(base_ring)
or number_field.all.is_NumberField(base_ring)
or ( sage.rings.fraction_field.is_FractionField(base_ring) and ( base_ring.base_ring().is_prime_field() or base_ring.base_ring() is ZZ ) )
or base_ring is ZZ
or is_IntegerModRing(base_ring) )示例2: is_locally_solvable
def is_locally_solvable(self, p):
r"""
Returns True if and only if ``self`` has a solution over the
`p`-adic numbers. Here `p` is a prime number or equals
`-1`, infinity, or `\RR` to denote the infinite place.
EXAMPLES::
sage: C = Conic(QQ, [1,2,3])
sage: C.is_locally_solvable(-1)
False
sage: C.is_locally_solvable(2)
False
sage: C.is_locally_solvable(3)
True
sage: C.is_locally_solvable(QQ.hom(RR))
False
sage: D = Conic(QQ, [1, 2, -3])
sage: D.is_locally_solvable(infinity)
True
sage: D.is_locally_solvable(RR)
True
"""
from sage.categories.map import Map
from sage.categories.all import Rings
D, T = self.diagonal_matrix()
abc = [D[j, j] for j in range(3)]
if abc[2] == 0:
return True
a = -abc[0]/abc[2]
b = -abc[1]/abc[2]
if is_RealField(p) or is_InfinityElement(p):
p = -1
elif isinstance(p, Map) and p.category_for().is_subcategory(Rings()):
# p is a morphism of Rings
if p.domain() is QQ and is_RealField(p.codomain()):
p = -1
else:
raise TypeError("p (=%s) needs to be a prime of base field " \
"B ( =`QQ`) in is_locally_solvable" % p)
if hilbert_symbol(a, b, p) == -1:
if self._local_obstruction is None:
self._local_obstruction = p
return False
return True示例3: can_convert_to_singular
def can_convert_to_singular(R):
"""
Returns True if this ring's base field or ring can be
represented in Singular, and the polynomial ring has at
least one generator. If this is True then this polynomial
ring can be represented in Singular.
The following base rings are supported: finite fields, rationals, number
fields, and real and complex fields.
EXAMPLES::
sage: from sage.rings.polynomial.polynomial_singular_interface import can_convert_to_singular
sage: can_convert_to_singular(PolynomialRing(QQ, names=['x']))
True
sage: can_convert_to_singular(PolynomialRing(QQ, names=[]))
False
TESTS:
Avoid non absolute number fields (see :trac:`23535`)::
sage: K.<a,b> = NumberField([x^2-2,x^2-5])
sage: can_convert_to_singular(K['s,t'])
False
"""
if R.ngens() == 0:
return False;
base_ring = R.base_ring()
if (base_ring is ZZ
or sage.rings.finite_rings.finite_field_constructor.is_FiniteField(base_ring)
or is_RationalField(base_ring)
or is_IntegerModRing(base_ring)
or is_RealField(base_ring)
or is_ComplexField(base_ring)
or is_RealDoubleField(base_ring)
or is_ComplexDoubleField(base_ring)):
return True
elif base_ring.is_prime_field():
return base_ring.characteristic() <= 2147483647
elif number_field.number_field_base.is_NumberField(base_ring):
return base_ring.is_absolute()
elif sage.rings.fraction_field.is_FractionField(base_ring):
B = base_ring.base_ring()
return B.is_prime_field() or B is ZZ or is_FiniteField(B)
elif is_RationalFunctionField(base_ring):
return base_ring.constant_field().is_prime_field()
else:
return False示例4: _evalf_
def _evalf_(self, n, x, **kwds):
"""
Evaluate :class:`chebyshev_U` numerically with mpmath.
EXAMPLES::
sage: chebyshev_U(5,-4+3.*I)
98280.0000000000 - 11310.0000000000*I
sage: chebyshev_U(10,3).n(75)
4.661117900000000000000e7
sage: chebyshev_U._evalf_(1.5, Mod(8,9))
Traceback (most recent call last):
...
TypeError: cannot evaluate chebyshev_U with parent Ring of integers modulo 9
"""
try:
real_parent = kwds["parent"]
except KeyError:
real_parent = parent(x)
if not is_RealField(real_parent) and not is_ComplexField(real_parent):
# parent is not a real or complex field: figure out a good parent
if x in RR:
x = RR(x)
real_parent = RR
elif x in CC:
x = CC(x)
real_parent = CC
if not is_RealField(real_parent) and not is_ComplexField(real_parent):
raise TypeError("cannot evaluate chebyshev_U with parent {}".format(real_parent))
from sage.libs.mpmath.all import call as mpcall
from sage.libs.mpmath.all import chebyu as mpchebyu
return mpcall(mpchebyu, n, x, parent=real_parent)示例5: quadratic_L_function__numerical
def quadratic_L_function__numerical(n, d, num_terms=1000):
"""
Evaluate the Dirichlet L-function (for quadratic character) numerically
(in a very naive way).
EXAMPLES:
First, let us test several values for a given character::
sage: RR = RealField(100)
sage: for i in range(5):
....: print("L({}, (-4/.)): {}".format(1+2*i, RR(quadratic_L_function__exact(1+2*i, -4)) - quadratic_L_function__numerical(RR(1+2*i),-4, 10000)))
L(1, (-4/.)): 0.000049999999500000024999996962707
L(3, (-4/.)): 4.99999970000003...e-13
L(5, (-4/.)): 4.99999922759382...e-21
L(7, (-4/.)): ...e-29
L(9, (-4/.)): ...e-29
This procedure fails for negative special values, as the Dirichlet
series does not converge here::
sage: quadratic_L_function__numerical(-3,-4, 10000)
Traceback (most recent call last):
...
ValueError: the Dirichlet series does not converge here
Test for several characters that the result agrees with the exact
value, to a given accuracy ::
sage: for d in range(-20,0): # long time (2s on sage.math 2014)
....: if abs(RR(quadratic_L_function__numerical(1, d, 10000) - quadratic_L_function__exact(1, d))) > 0.001:
....: print("Oops! We have a problem at d = {}: exact = {}, numerical = {}".format(d, RR(quadratic_L_function__exact(1, d)), RR(quadratic_L_function__numerical(1, d))))
"""
# Set the correct precision if it is given (for n).
if is_RealField(n.parent()):
R = n.parent()
else:
R = RealField()
if n < 0:
raise ValueError('the Dirichlet series does not converge here')
d1 = fundamental_discriminant(d)
ans = R.zero()
for i in range(1,num_terms):
ans += R(kronecker_symbol(d1,i) / R(i)**n)
return ans示例6: __init__
def __init__(self, X, P, codomain = None, check = False):
r"""
Create the discrete probability space with probabilities on the
space X given by the dictionary P with values in the field
real_field.
EXAMPLES::
sage: S = [ i for i in range(16) ]
sage: P = {}
sage: for i in range(15): P[i] = 2^(-i-1)
sage: P[15] = 2^-16
sage: X = DiscreteProbabilitySpace(S,P)
sage: X.domain()
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15)
sage: X.set()
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
sage: X.entropy()
1.9997253418
A probability space can be defined on any list of elements.
EXAMPLES::
sage: AZ = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'
sage: S = [ AZ[i] for i in range(26) ]
sage: P = { 'A':1/2, 'B':1/4, 'C':1/4 }
sage: X = DiscreteProbabilitySpace(S,P)
sage: X
Discrete probability space defined by {'A': 1/2, 'C': 1/4, 'B': 1/4}
sage: X.entropy()
1.5
"""
if codomain is None:
codomain = RealField()
if not is_RealField(codomain) and not is_RationalField(codomain):
raise TypeError("Argument codomain (= %s) must be the reals or rationals" % codomain)
if check:
one = sum([ P[x] for x in P.keys() ])
if is_RationalField(codomain):
if not one == 1:
raise TypeError("Argument P (= %s) does not define a probability function")
else:
if not Abs(one-1) < 2^(-codomain.precision()+1):
raise TypeError("Argument P (= %s) does not define a probability function")
ProbabilitySpace_generic.__init__(self, X, codomain)
DiscreteRandomVariable.__init__(self, self, P, codomain, check)示例7: complex_conjugation
def complex_conjugation(self, P=None):
"""
Return the unique element of self corresponding to complex conjugation,
for a specified embedding P into the complex numbers. If P is not
specified, use the "standard" embedding, whenever that is well-defined.
EXAMPLE::
sage: L = CyclotomicField(7)
sage: G = L.galois_group()
sage: G.complex_conjugation()
(1,6)(2,3)(4,5)
An example where the field is not CM, so complex conjugation really
depends on the choice of embedding::
sage: L = NumberField(x^6 + 40*x^3 + 1372,'a')
sage: G = L.galois_group()
sage: [G.complex_conjugation(x) for x in L.places()]
[(1,3)(2,6)(4,5), (1,5)(2,4)(3,6), (1,2)(3,4)(5,6)]
"""
if P is None:
Q = self.splitting_field().specified_complex_embedding()
if Q is None:
raise ValueError, "No default complex embedding specified"
P = Q
from sage.rings.real_mpfr import is_RealField
from sage.rings.qqbar import is_AlgebraicRealField
if is_RealField(P.codomain()) or is_AlgebraicRealField(P.codomain()):
return self.one()
P = refine_embedding(P, infinity)
if self._non_galois_not_impl:
raise TypeError, "Extension is not Galois"
g = self.splitting_field().gen()
gconj = P(g).conjugate()
elts = [s for s in self if P(s(g)) == gconj]
if len(elts) != 1: raise ArithmeticError, "Something has gone very wrong here"
return elts[0]示例8: quadratic_L_function__numerical
def quadratic_L_function__numerical(n, d, num_terms=1000):
"""
Evaluate the Dirichlet L-function (for quadratic character) numerically
(in a very naive way).
EXAMPLES::
sage: ## Test several values for a given character
sage: RR = RealField(100)
sage: for i in range(5):
... print "L(" + str(1+2*i) + ", (-4/.)): ", RR(quadratic_L_function__exact(1+2*i, -4)) - quadratic_L_function__numerical(RR(1+2*i),-4, 10000)
L(1, (-4/.)): 0.000049999999500000024999996962707
L(3, (-4/.)): 4.99999970000003...e-13
L(5, (-4/.)): 4.99999922759382...e-21
L(7, (-4/.)): ...e-29
L(9, (-4/.)): ...e-29
sage: ## Testing the accuracy of the negative special values
sage: ## ---- THIS FAILS SINCE THE DIRICHLET SERIES DOESN'T CONVERGE HERE! ----
sage: ## Test several characters agree with the exact value, to a given accuracy.
sage: for d in range(-20,0):
... if abs(RR(quadratic_L_function__numerical(1, d, 10000) - quadratic_L_function__exact(1, d))) > 0.001:
... print "Oops! We have a problem at d = ", d, " exact = ", RR(quadratic_L_function__exact(1, d)), " numerical = ", RR(quadratic_L_function__numerical(1, d))
...
"""
## Set the correct precision if it's given (for n).
if is_RealField(n.parent()):
R = n.parent()
else:
R = RealField()
d1 = fundamental_discriminant(d)
ans = R(0)
for i in range(1,num_terms):
ans += R(kronecker_symbol(d1,i) / R(i)**n)
return ans示例9: isogenies_prime_degree
#.........这里部分代码省略.........
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 18.0000000000000 over Complex Field with 53 bits of precision
sage: E.isogenies_prime_degree(11)
Traceback (most recent call last):
...
NotImplementedError: This code could be implemented for general complex fields, but has not been yet.
Examples over finite fields::
sage: E = EllipticCurve(GF(next_prime(1000000)), [7,8])
sage: E.isogenies_prime_degree()
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 347438*x + 594729 over Finite Field of size 1000003, Isogeny of degree 17 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 674846*x + 7392 over Finite Field of size 1000003, Isogeny of degree 23 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 390065*x + 605596 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(5)
[]
sage: E.isogenies_prime_degree(7)
[]
sage: E.isogenies_prime_degree(13)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003,
Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 3, 5, 7, 13])
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
sage: E.isogenies_prime_degree([2, 4])
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree(4)
Traceback (most recent call last):
...
ValueError: 4 is not prime.
sage: E.isogenies_prime_degree(11)
[]
sage: E = EllipticCurve(GF(17),[2,0])
sage: E.isogenies_prime_degree(3)
[]
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 9*x over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 9 over Finite Field of size 17, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 17 to Elliptic Curve defined by y^2 = x^3 + 5*x + 8 over Finite Field of size 17]
sage: E = EllipticCurve(GF(13^4, 'a'),[2,8])
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 7*x + 4 over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (8*a^3+2*a^2+7*a+5)*x + (12*a^3+3*a^2+4*a+4) over Finite Field in a of size 13^4, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + (5*a^3+11*a^2+6*a+11)*x + (a^3+10*a^2+9*a) over Finite Field in a of size 13^4]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 9*x + 11 over Finite Field in a of size 13^4]
Example to show that separable isogenies of degree equal to the characteristic are now implemented::
sage: E.isogenies_prime_degree(13)
[Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field in a of size 13^4 to Elliptic Curve defined by y^2 = x^3 + 6*x + 5 over Finite Field in a of size 13^4]
Examples over number fields (other than QQ)::
sage: QQroot2.<e> = NumberField(x^2-2)
sage: E = EllipticCurve(QQroot2, j=8000)
sage: E.isogenies_prime_degree()
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (-602112000)*x + 5035261952000 over Number Field in e with defining polynomial x^2 - 2,
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (903168000*e-1053696000)*x + (14161674240000*e-23288086528000) over Number Field in e with defining polynomial x^2 - 2,
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-150528000)*x + (-629407744000) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 = x^3 + (-903168000*e-1053696000)*x + (-14161674240000*e-23288086528000) over Number Field in e with defining polynomial x^2 - 2]
sage: E = EllipticCurve(QQroot2, [1,0,1,4, -6]); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2
sage: E.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-36)*x + (-70) over Number Field in e with defining polynomial x^2 - 2]
sage: E.isogenies_prime_degree(3)
[Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-128/3)*x + 5662/27 over Number Field in e with defining polynomial x^2 - 2, Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-171)*x + (-874) over Number Field in e with defining polynomial x^2 - 2]
"""
F = self.base_ring()
if is_RealField(F):
raise NotImplementedError("This code could be implemented for general real fields, but has not been yet.")
if is_ComplexField(F):
raise NotImplementedError("This code could be implemented for general complex fields, but has not been yet.")
if F == rings.QQbar:
raise NotImplementedError("This code could be implemented for QQbar, but has not been yet.")
from isogeny_small_degree import isogenies_prime_degree
if l is None:
from sage.rings.all import prime_range
l = prime_range(max_l+1)
if not isinstance(l, list):
try:
l = rings.ZZ(l)
except TypeError:
raise ValueError("%s is not prime."%l)
if l.is_prime():
return isogenies_prime_degree(self, l)
else:
raise ValueError("%s is not prime."%l)
L = list(set(l))
try:
L = [rings.ZZ(l) for l in L]
except TypeError:
raise ValueError("%s is not a list of primes."%l)
L.sort()
return sum([isogenies_prime_degree(self,l) for l in L],[])示例10: has_rational_point
#.........这里部分代码省略.........
sage: K.coerce_embedding()
Generic morphism:
From: Number Field in i with defining polynomial x^2 + 1
To: Complex Lazy Field
Defn: i -> 1*I
sage: Conic(K, [1,1,1]).rational_point(algorithm='magma') # optional - magma
(-i : 1 : 0)
sage: x = QQ['x'].gen()
sage: L.<i> = NumberField(x^2+1, embedding=None)
sage: Conic(L, [1,1,1]).rational_point(algorithm='magma') # optional - magma
(-i : 1 : 0)
sage: L == K
False
"""
if read_cache:
if self._rational_point is not None:
if point:
return True, self._rational_point
else:
return True
B = self.base_ring()
if algorithm == 'magma':
from sage.interfaces.magma import magma
M = magma(self)
b = M.HasRationalPoint().sage()
if not point:
return b
if not b:
return False, None
M_pt = M.HasRationalPoint(nvals=2)[1]
# Various attempts will be made to convert `pt` to
# a Sage object. The end result will always be checked
# by self.point().
pt = [M_pt[1], M_pt[2], M_pt[3]]
# The first attempt is to use sequences. This is efficient and
# succeeds in cases where the Magma interface fails to convert
# number field elements, because embeddings between number fields
# may be lost on conversion to and from Magma.
# This should deal with all absolute number fields.
try:
return True, self.point([B(c.Eltseq().sage()) for c in pt])
except TypeError:
pass
# The second attempt tries to split Magma elements into
# numerators and denominators first. This is neccessary
# for the field of rational functions, because (at the moment of
# writing) fraction field elements are not converted automatically
# from Magma to Sage.
try:
return True, self.point( \
[B(c.Numerator().sage()/c.Denominator().sage()) for c in pt])
except (TypeError, NameError):
pass
# Finally, let the Magma interface handle conversion.
try:
return True, self.point([B(c.sage()) for c in pt])
except (TypeError, NameError):
pass
raise NotImplementedError("No correct conversion implemented for converting the Magma point %s on %s to a correct Sage point on self (=%s)" % (M_pt, M, self))
if algorithm != 'default':
raise ValueError("Unknown algorithm: %s" % algorithm)
if is_ComplexField(B):
if point:
[_,_,_,d,e,f] = self._coefficients
if d == 0:
return True, self.point([0,1,0])
return True, self.point([0, ((e**2-4*d*f).sqrt()-e)/(2*d), 1],
check = False)
return True
if is_RealField(B):
D, T = self.diagonal_matrix()
[a, b, c] = [D[0,0], D[1,1], D[2,2]]
if a == 0:
ret = True, self.point(T*vector([1,0,0]), check = False)
elif a*c <= 0:
ret = True, self.point(T*vector([(-c/a).sqrt(),0,1]),
check = False)
elif b == 0:
ret = True, self.point(T*vector([0,1,0]), check = False)
elif b*c <= 0:
ret = True, self.point(T*vector([0,(-c/b).sqrt(),0,1]),
check = False)
else:
ret = False, None
if point:
return ret
return ret[0]
raise NotImplementedError("has_rational_point not implemented for " \
"conics over base field %s" % B)示例11: _singular_init_
def _singular_init_(self, singular=singular_default):
"""
Return a newly created Singular ring matching this ring.
EXAMPLES::
sage: PolynomialRing(QQ,'u_ba')._singular_init_()
// characteristic : 0
// number of vars : 1
// block 1 : ordering lp
// : names u_ba
// block 2 : ordering C
"""
if not can_convert_to_singular(self):
raise TypeError, "no conversion of this ring to a Singular ring defined"
if self.ngens()==1:
_vars = '(%s)'%self.gen()
if "*" in _vars: # 1.000...000*x
_vars = _vars.split("*")[1]
order = 'lp'
else:
_vars = str(self.gens())
order = self.term_order().singular_str()
base_ring = self.base_ring()
if is_RealField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
precision = base_ring.precision()
digits = sage.rings.arith.integer_ceil((2*precision - 2)/7.0)
self.__singular = singular.ring("(real,%d,0)"%digits, _vars, order=order, check=False)
elif is_ComplexField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
precision = base_ring.precision()
digits = sage.rings.arith.integer_ceil((2*precision - 2)/7.0)
self.__singular = singular.ring("(complex,%d,0,I)"%digits, _vars, order=order, check=False)
elif is_RealDoubleField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
self.__singular = singular.ring("(real,15,0)", _vars, order=order, check=False)
elif is_ComplexDoubleField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
self.__singular = singular.ring("(complex,15,0,I)", _vars, order=order, check=False)
elif base_ring.is_prime_field():
self.__singular = singular.ring(self.characteristic(), _vars, order=order, check=False)
elif sage.rings.finite_rings.constructor.is_FiniteField(base_ring):
# not the prime field!
gen = str(base_ring.gen())
r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order, check=False)
self.__minpoly = (str(base_ring.modulus()).replace("x",gen)).replace(" ","")
if singular.eval('minpoly') != "(" + self.__minpoly + ")":
singular.eval("minpoly=%s"%(self.__minpoly) )
self.__minpoly = singular.eval('minpoly')[1:-1]
self.__singular = r
elif number_field.all.is_NumberField(base_ring) and base_ring.is_absolute():
# not the rationals!
gen = str(base_ring.gen())
poly=base_ring.polynomial()
poly_gen=str(poly.parent().gen())
poly_str=str(poly).replace(poly_gen,gen)
r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order, check=False)
self.__minpoly = (poly_str).replace(" ","")
if singular.eval('minpoly') != "(" + self.__minpoly + ")":
singular.eval("minpoly=%s"%(self.__minpoly) )
self.__minpoly = singular.eval('minpoly')[1:-1]
self.__singular = r
elif sage.rings.fraction_field.is_FractionField(base_ring) and (base_ring.base_ring() is ZZ or base_ring.base_ring().is_prime_field()):
if base_ring.ngens()==1:
gens = str(base_ring.gen())
else:
gens = str(base_ring.gens())
self.__singular = singular.ring( "(%s,%s)"%(base_ring.characteristic(),gens), _vars, order=order, check=False)
elif is_IntegerModRing(base_ring):
ch = base_ring.characteristic()
if ch.is_power_of(2):
exp = ch.nbits() -1
self.__singular = singular.ring("(integer,2,%d)"%(exp,), _vars, order=order, check=False)
else:
self.__singular = singular.ring("(integer,%d)"%(ch,), _vars, order=order, check=False)
elif base_ring is ZZ:
self.__singular = singular.ring("(integer)", _vars, order=order, check=False)
else:
raise TypeError, "no conversion to a Singular ring defined"
return self.__singular示例12: canonical_height
def canonical_height(self, F, **kwds):
r"""
Evaluates the (absolute) canonical height of ``self`` with respect to ``F``. Must be over number field
or order of a number field. Specify either the number of terms of the series to evaluate or
the error bound required.
ALGORITHM:
The sum of the Green's function at the archimedean places and the places of bad reduction.
INPUT:
- ``F`` - a projective morphism
kwds:
- ``badprimes`` - a list of primes of bad reduction
- ``N`` - positive integer. number of terms of the series to use in the local green functions
- ``prec`` - positive integer, float point or p-adic precision
- ``error_bound`` - a positive real number
OUTPUT:
- a real number
EXAMPLES::
sage: P.<x,y> = ProjectiveSpace(ZZ,1)
sage: H = Hom(P,P)
sage: f = H([x^2+y^2,2*x*y]);
sage: Q = P(2,1)
sage: f.canonical_height(f(Q))
2.1965476757927038111992627081
sage: f.canonical_height(Q)
1.0979353871245941198040174712
Notice that preperiodic points may not be exactly 0. ::
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(P,P)
sage: f = H([x^2-29/16*y^2,y^2]);
sage: Q = P(5,4)
sage: f.canonical_height(Q, N=30)
1.4989058602918874235833076226e-9
::
sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: X = P.subscheme(x^2-y^2);
sage: H = Hom(X,X)
sage: f = H([x^2,y^2,30*z^2]);
sage: Q = X([4,4,1])
sage: f.canonical_height(Q, badprimes=[2,3,5], prec=200)
2.7054056208276961889784303469356774912979228770208655455481
"""
bad_primes = kwds.pop("badprimes", None)
prec = kwds.get("prec", 100)
error_bound = kwds.get("error_bound", None)
K = FractionField(self.codomain().base_ring())
if not K in _NumberFields:
raise NotImplementedError("Must be over a NumberField or a NumberField Order")
if bad_primes is None:
bad_primes = []
for b in self:
if K == QQ:
bad_primes += b.denominator().prime_factors()
else:
bad_primes += b.denominator_ideal().prime_factors()
bad_primes += K(F.resultant()).support()
bad_primes = list(set(bad_primes))
emb = K.places(prec=prec)
num_places = len(emb) + len(bad_primes)
if not error_bound is None:
error_bound /= num_places
R = RealField(prec)
h = R(0)
# Archimedean local heights
# :: WARNING: If places is fed the default Sage precision of 53 bits,
# it uses Real or Complex Double Field in place of RealField(prec) or ComplexField(prec)
# the function is_RealField does not identify RDF as real, so we test for that ourselves.
for v in emb:
if is_RealField(v.codomain()) or v.codomain() is RDF:
dv = R(1)
else:
dv = R(2)
h += dv*self.green_function(F, v, **kwds) #arch Green function
# Non-Archimedean local heights
for v in bad_primes:
if K == QQ:
dv = R(1)
else:
dv = R(v.residue_class_degree() * v.absolute_ramification_index())
#.........这里部分代码省略.........
示例13: _singular_init_
def _singular_init_(self, singular=singular):
"""
Return a newly created Singular ring matching this ring.
EXAMPLES::
sage: PolynomialRing(QQ,'u_ba')._singular_init_()
polynomial ring, over a field, global ordering
// coefficients: QQ
// number of vars : 1
// block 1 : ordering lp
// : names u_ba
// block 2 : ordering C
"""
if not can_convert_to_singular(self):
raise TypeError("no conversion of this ring to a Singular ring defined")
if self.ngens()==1:
_vars = '(%s)'%self.gen()
if "*" in _vars: # 1.000...000*x
_vars = _vars.split("*")[1]
order = 'lp'
else:
_vars = str(self.gens())
order = self.term_order().singular_str()
base_ring = self.base_ring()
if is_RealField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
precision = base_ring.precision()
digits = sage.arith.all.integer_ceil((2*precision - 2)/7.0)
self.__singular = singular.ring("(real,%d,0)"%digits, _vars, order=order, check=False)
elif is_ComplexField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
precision = base_ring.precision()
digits = sage.arith.all.integer_ceil((2*precision - 2)/7.0)
self.__singular = singular.ring("(complex,%d,0,I)"%digits, _vars, order=order, check=False)
elif is_RealDoubleField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
self.__singular = singular.ring("(real,15,0)", _vars, order=order, check=False)
elif is_ComplexDoubleField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
self.__singular = singular.ring("(complex,15,0,I)", _vars, order=order, check=False)
elif base_ring.is_prime_field():
self.__singular = singular.ring(self.characteristic(), _vars, order=order, check=False)
elif sage.rings.finite_rings.finite_field_constructor.is_FiniteField(base_ring):
# not the prime field!
gen = str(base_ring.gen())
r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order, check=False)
self.__minpoly = (str(base_ring.modulus()).replace("x",gen)).replace(" ","")
if singular.eval('minpoly') != "(" + self.__minpoly + ")":
singular.eval("minpoly=%s"%(self.__minpoly) )
self.__minpoly = singular.eval('minpoly')[1:-1]
self.__singular = r
elif number_field.number_field_base.is_NumberField(base_ring) and base_ring.is_absolute():
# not the rationals!
gen = str(base_ring.gen())
poly=base_ring.polynomial()
poly_gen=str(poly.parent().gen())
poly_str=str(poly).replace(poly_gen,gen)
r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order, check=False)
self.__minpoly = (poly_str).replace(" ","")
if singular.eval('minpoly') != "(" + self.__minpoly + ")":
singular.eval("minpoly=%s"%(self.__minpoly) )
self.__minpoly = singular.eval('minpoly')[1:-1]
self.__singular = r
elif sage.rings.fraction_field.is_FractionField(base_ring) and (base_ring.base_ring() is ZZ or base_ring.base_ring().is_prime_field() or is_FiniteField(base_ring.base_ring())):
if base_ring.ngens()==1:
gens = str(base_ring.gen())
else:
gens = str(base_ring.gens())
if not (not base_ring.base_ring().is_prime_field() and is_FiniteField(base_ring.base_ring())) :
self.__singular = singular.ring( "(%s,%s)"%(base_ring.characteristic(),gens), _vars, order=order, check=False)
else:
ext_gen = str(base_ring.base_ring().gen())
_vars = '(' + ext_gen + ', ' + _vars[1:];
R = self.__singular = singular.ring( "(%s,%s)"%(base_ring.characteristic(),gens), _vars, order=order, check=False)
self.base_ring().__minpoly = (str(base_ring.base_ring().modulus()).replace("x",ext_gen)).replace(" ","")
singular.eval('setring '+R._name);
from sage.misc.stopgap import stopgap
stopgap("Denominators of fraction field elements are sometimes dropped without warning.", 17696)
#.........这里部分代码省略.........