def __init__(self, expo, values, dop=None, check=True):
r"""
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.naive_sum import *
sage: Dops, x, Dx = DifferentialOperators()
sage: LogSeriesInitialValues(0, {0: (1, 0)}, x*Dx^3 + 2*Dx^2 + x*Dx)
Traceback (most recent call last):
...
ValueError: invalid initial data for x*Dx^3 + 2*Dx^2 + x*Dx at 0
"""
try:
self.expo = QQ.coerce(expo)
except TypeError:
self.expo = QQbar.coerce(expo)
if isinstance(values, dict):
all_values = sum(values.values(), ()) # concatenation of tuples
else:
all_values = values
values = dict((n, (values[n], )) for n in xrange(len(values)))
self.universe = Sequence(all_values).universe()
if not utilities.is_numeric_parent(self.universe):
raise ValueError("initial values must coerce into a ball field")
self.shift = {
s: tuple(self.universe(a) for a in ini)
for s, ini in values.iteritems()
}
try:
if check and dop is not None and not self.is_valid_for(dop):
raise ValueError(
"invalid initial data for {} at 0".format(dop))
except TypeError: # coercion problems btw QQbar and number fields
pass
def __init__(self, expo, values, dop=None, check=True, mults=None):
r"""
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.naive_sum import *
sage: from ore_algebra.analytic.differential_operator import DifferentialOperator
sage: Dops, x, Dx = DifferentialOperators()
sage: LogSeriesInitialValues(0, {0: (1, 0)},
....: DifferentialOperator(x*Dx^3 + 2*Dx^2 + x*Dx))
Traceback (most recent call last):
...
ValueError: invalid initial data for x*Dx^3 + 2*Dx^2 + x*Dx at 0
"""
try:
self.expo = QQ.coerce(expo)
except TypeError:
try:
self.expo = QQbar.coerce(expo)
except TypeError:
# symbolic; won't be sortable
self.expo = expo
if isinstance(values, dict):
all_values = tuple(
chain.from_iterable(ini if isinstance(ini, tuple) else (ini, )
for ini in values.values()))
else:
all_values = values
values = dict((n, (values[n], )) for n in range(len(values)))
self.universe = Sequence(all_values).universe()
if not utilities.is_numeric_parent(self.universe):
raise ValueError("initial values must coerce into a ball field")
self.shift = {}
if mults is not None:
for s, m in mults:
self.shift[s] = [self.universe.zero()] * m
for k, ini in values.items():
if isinstance(k, tuple): # requires mult != None
s, m = k
s = int(s)
self.shift[s][m] = self.universe(ini)
else:
s = int(k)
self.shift[s] = tuple(self.universe(a) for a in ini)
self.shift = {s: tuple(ini) for s, ini in self.shift.items()}
try:
if check and dop is not None and not self.is_valid_for(dop):
raise ValueError(
"invalid initial data for {} at 0".format(dop))
except TypeError: # coercion problems btw QQbar and number fields
pass
def as_embedded_number_field_element(alg):
from sage.rings.number_field.number_field import NumberField
nf, elt, emb = alg.as_number_field_element()
if nf is QQ:
res = elt
else:
embnf = NumberField(nf.polynomial(),
nf.variable_name(),
embedding=emb(nf.gen()))
res = elt.polynomial()(embnf.gen())
assert QQbar.coerce(res) == alg
return res
def as_embedded_number_field_elements(algs):
try:
nf, elts, _ = number_field_elements_from_algebraics(algs, embedded=True)
except NotImplementedError: # compatibility with Sage <= 9.3
nf, elts, emb = number_field_elements_from_algebraics(algs)
if nf is not QQ:
nf = NumberField(nf.polynomial(), nf.variable_name(),
embedding=emb(nf.gen()))
elts = [elt.polynomial()(nf.gen()) for elt in elts]
nf, hom = good_number_field(nf)
elts = [hom(elt) for elt in elts]
assert [QQbar.coerce(elt) == alg for alg, elt in zip(algs, elts)]
return nf, elts
def _try_merge_conjugate_singularities(dop, sing, base, todo):
if any(c not in QQ for pol in dop for c in pol):
return False
need_conjugates = False
sgn = 1 if QQbar.coerce(base.value).imag() >= 0 else -1
for x in sing:
if sgn * x.imag() < 0:
need_conjugates = True
del todo[x]
xconj = x.conjugate()
if xconj not in todo:
todo[xconj] = LocalMonodromyData()
todo[xconj].want_conj = True
return need_conjugates
def is_ordinary(self):
if self._force_singular:
return False
lc = self.dop.leading_coefficient()
if not lc(self.iv()).contains_zero():
return True
if self.is_exact():
try:
val = lc(self.value)
except TypeError: # work around coercion weaknesses
val = lc.change_ring(QQbar)(QQbar.coerce(self.value))
return not val.is_zero()
else:
raise ValueError("can't tell if inexact point is singular")
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 _monodromy_matrices(dop, base, eps=1e-16, sing=None):
r"""
EXAMPLES::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.monodromy import _monodromy_matrices
sage: Dops, x, Dx = DifferentialOperators()
sage: rat = 1/(x^2-1)
sage: dop = (rat*Dx - rat.derivative()).lclm(Dx*x*Dx)
sage: [rec.point for rec in _monodromy_matrices(dop, 0) if not rec.is_scalar]
[0]
TESTS::
sage: from ore_algebra.examples import fcc
sage: mon = list(_monodromy_matrices(fcc.dop5, -1, 2**(-2**7))) # long time (2.3 s)
sage: [rec.monodromy[0][0] for rec in mon if rec.point == -5/3] # long time
[[1.01088578589319884254557667137848...]]
Thanks to Alexandre Goyer for this example::
sage: L1 = ((x^5 - x^4 + x^3)*Dx^3 + (27/8*x^4 - 25/9*x^3 + 8*x^2)*Dx^2
....: + (37/24*x^3 - 25/9*x^2 + 14*x)*Dx - 2*x^2 - 3/4*x + 4)
sage: L2 = ((x^5 - 9/4*x^4 + x^3)*Dx^3 + (11/6*x^4 - 31/4*x^3 + 7*x^2)*Dx^2
....: + (7/30*x^3 - 101/20*x^2 + 10*x)*Dx + 4/5*x^2 + 5/6*x + 2)
sage: L = L1*L2
sage: L = L.parent()(L.annihilator_of_composition(x+1))
sage: mon = list(_monodromy_matrices(L, 0, eps=1e-30)) # long time (1.3-1.7 s)
sage: mon[-1][0], mon[-1][1][0][0] # long time
(0.6403882032022075?,
[1.15462187280628880820271...] + [-0.018967673022432256251718...]*I)
"""
dop = DifferentialOperator(dop)
base = QQbar.coerce(base)
eps = RBF(eps)
if sing is None:
sing = dop._singularities(QQbar)
else:
sing = [QQbar.coerce(s) for s in sing]
todo = {x: TodoItem(x, dop, want_self=True, want_conj=False) for x in sing}
base = todo.setdefault(base, TodoItem(base, dop))
if not base.point().is_regular():
raise ValueError("irregular singular base point")
# If the coefficients are rational, reduce to handling singularities in the
# same half-plane as the base point, and share some computations between
# Galois conjugates.
need_conjugates = False
crit_cache = None
if all(c in QQ for pol in dop for c in pol):
need_conjugates = _merge_conjugate_singularities(dop, sing, base, todo)
# TODO: do something like that even over number fields?
# XXX this is actually a bit costly: do it only after checking that the
# monodromy is not scalar?
# XXX keep the cache from one run to the next when increasing prec?
crit_cache = {}
Scalars = ComplexBallField(utilities.prec_from_eps(eps))
id_mat = matrix.identity_matrix(Scalars, dop.order())
def matprod(elts):
return prod(reversed(elts), id_mat)
for key, todoitem in list(todo.items()):
point = todoitem.point()
# We could call _local_monodromy_loop() if point is irregular, but
# delaying it may allow us to start returning results earlier.
if point.is_regular():
if crit_cache is None or point.algdeg() == 1:
crit = _critical_monomials(dop.shift(point))
emb = point.value.parent().hom(Scalars)
else:
mpol = point.value.minpoly()
try:
NF, crit = crit_cache[mpol]
except KeyError:
NF = point.value.parent()
crit = _critical_monomials(dop.shift(point))
# Only store the critical monomials for reusing when all
# local exponents are rational. We need to restrict to this
# case because we do not have the technology in place to
# follow algebraic exponents along the embedding of NF in ℂ.
# (They are represented as elements of "new" number fields
# given by as_embedded_number_field_element(), even when
# they actually lie in NF itself as opposed to a further
# algebraic extension. XXX: Ideally, LocalBasisMapper should
# give us access to the tower of extensions in which the
# exponents "naturally" live.)
if all(sol.leftmost.parent() is QQ for sol in crit):
crit_cache[mpol] = NF, crit
emb = NF.hom([Scalars(point.value.parent().gen())],
check=False)
mon, scalar = _formal_monodromy_from_critical_monomials(crit, emb)
if scalar:
# No need to compute the connection matrices then!
# XXX When we do need them, though, it would be better to get
# the formal monodromy as a byproduct of their computation.
if todoitem.want_self:
yield LocalMonodromyData(key, mon, True)
if todoitem.want_conj:
conj = key.conjugate()
logger.info(
"Computing local monodromy around %s by "
"complex conjugation", conj)
conj_mat = ~mon.conjugate()
yield LocalMonodromyData(conj, conj_mat, True)
if todoitem is not base:
del todo[key]
continue
todoitem.local_monodromy = [mon]
todoitem.polygon = [point]
if need_conjugates:
base_conj_mat = dop.numerical_transition_matrix(
[base.point(), base.point().conjugate()],
eps,
assume_analytic=True)
def conjugate_monodromy(mat):
return ~base_conj_mat * ~mat.conjugate() * base_conj_mat
tree = _spanning_tree(base, todo.values())
def dfs(x, path, path_mat):
logger.info("Computing local monodromy around %s via %s", x, path)
local_mat = matprod(x.local_monodromy)
based_mat = (~path_mat) * local_mat * path_mat
if x.want_self:
yield LocalMonodromyData(x.alg, based_mat, False)
if x.want_conj:
conj = x.alg.conjugate()
logger.info(
"Computing local monodromy around %s by complex "
"conjugation", conj)
conj_mat = conjugate_monodromy(based_mat)
yield LocalMonodromyData(conj, conj_mat, False)
x.done = True
for y in tree.neighbors(x):
if y.done:
continue
if y.local_monodromy is None:
y.polygon, y.local_monodromy = _local_monodromy_loop(
dop, y.point(), eps)
new_path_mat = _extend_path_mat(dop, path_mat, x, y, eps, matprod)
yield from dfs(y, path + [y], new_path_mat)
yield from dfs(base, [base], id_mat)
def _monodromy_matrices(dop, base, eps=1e-16, sing=None):
r"""
EXAMPLES::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.monodromy import _monodromy_matrices
sage: Dops, x, Dx = DifferentialOperators()
sage: rat = 1/(x^2-1)
sage: dop = (rat*Dx - rat.derivative()).lclm(Dx*x*Dx)
sage: [rec.point for rec in _monodromy_matrices(dop, 0) if not rec.is_scalar]
[0]
TESTS::
sage: from ore_algebra.examples import fcc
sage: mon = list(_monodromy_matrices(fcc.dop5, -1, 2**(-2**7))) # long time (2.3 s)
sage: [rec.monodromy[0][0] for rec in mon if rec.point == -5/3] # long time
[[1.01088578589319884254557667137848...]]
"""
dop = DifferentialOperator(dop)
base = QQbar.coerce(base)
eps = RBF(eps)
if sing is None:
sing = dop._singularities(QQbar)
else:
sing = [QQbar.coerce(s) for s in sing]
todo = {x: TodoItem(x, dop, want_self=True, want_conj=False) for x in sing}
base = todo.setdefault(base, TodoItem(base, dop))
if not base.point().is_regular():
raise ValueError("irregular singular base point")
# If the coefficients are rational, reduce to handling singularities in the
# same half-plane as the base point, and share some computations between
# Galois conjugates.
need_conjugates = False
crit_cache = None
if all(c in QQ for pol in dop for c in pol):
need_conjugates = _merge_conjugate_singularities(dop, sing, base, todo)
# TODO: do something like that even over number fields?
# XXX this is actually a bit costly: do it only after checking that the
# monodromy is not scalar?
# XXX keep the cache from one run to the next when increasing prec?
crit_cache = {}
Scalars = ComplexBallField(utilities.prec_from_eps(eps))
id_mat = matrix.identity_matrix(Scalars, dop.order())
def matprod(elts):
return prod(reversed(elts), id_mat)
for key, todoitem in list(todo.items()):
point = todoitem.point()
# We could call _local_monodromy_loop() if point is irregular, but
# delaying it may allow us to start returning results earlier.
if point.is_regular():
if crit_cache is None or point.algdeg() == 1:
crit = _critical_monomials(dop.shift(point))
emb = point.value.parent().hom(Scalars)
else:
mpol = point.value.minpoly()
try:
NF, crit = crit_cache[mpol]
except KeyError:
NF = point.value.parent()
crit = _critical_monomials(dop.shift(point))
crit_cache[mpol] = NF, crit
emb = NF.hom([Scalars(point.value.parent().gen())],
check=False)
mon, scalar = _formal_monodromy_from_critical_monomials(crit, emb)
if scalar:
# No need to compute the connection matrices then!
# XXX When we do need them, though, it would be better to get
# the formal monodromy as a byproduct of their computation.
if todoitem.want_self:
yield LocalMonodromyData(key, mon, True)
if todoitem.want_conj:
conj = key.conjugate()
logger.info(
"Computing local monodromy around %s by "
"complex conjugation", conj)
conj_mat = ~mon.conjugate()
yield LocalMonodromyData(conj, conj_mat, True)
if todoitem is not base:
del todo[key]
continue
todoitem.local_monodromy = [mon]
todoitem.polygon = [point]
if need_conjugates:
base_conj_mat = dop.numerical_transition_matrix(
[base.point(), base.point().conjugate()],
eps,
assume_analytic=True)
def conjugate_monodromy(mat):
return ~base_conj_mat * ~mat.conjugate() * base_conj_mat
tree = _spanning_tree(base, todo.values())
def dfs(x, path, path_mat):
logger.info("Computing local monodromy around %s via %s", x, path)
local_mat = matprod(x.local_monodromy)
based_mat = (~path_mat) * local_mat * path_mat
if x.want_self:
yield LocalMonodromyData(x.alg, based_mat, False)
if x.want_conj:
conj = x.alg.conjugate()
logger.info(
"Computing local monodromy around %s by complex "
"conjugation", conj)
conj_mat = conjugate_monodromy(based_mat)
yield LocalMonodromyData(conj, conj_mat, False)
x.done = True
for y in tree.neighbors(x):
if y.done:
continue
if y.local_monodromy is None:
y.polygon, y.local_monodromy = _local_monodromy_loop(
dop, y.point(), eps)
new_path_mat = _extend_path_mat(dop, path_mat, x, y, eps, matprod)
yield from dfs(y, path + [y], new_path_mat)
yield from dfs(base, [base], id_mat)
def _monodromy_matrices(dop, base, eps=1e-16, sing=None):
r"""
EXAMPLES::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.monodromy import _monodromy_matrices
sage: Dops, x, Dx = DifferentialOperators()
sage: rat = 1/(x^2-1)
sage: dop = (rat*Dx - rat.derivative()).lclm(Dx*x*Dx)
sage: [rec.point for rec in _monodromy_matrices(dop, 0) if not rec.is_scalar]
[0]
"""
dop = DifferentialOperator(dop)
base = QQbar.coerce(base)
eps = RBF(eps)
if sing is None:
sing = dop._singularities(QQbar)
else:
sing = [QQbar.coerce(s) for s in sing]
todo = {
x: PointWithMonodromyData(x, dop, want_self=True, want_conj=False)
for x in sing
}
base = todo.setdefault(base, PointWithMonodromyData(base, dop))
if not base.is_regular():
raise ValueError("irregular singular base point")
# If the coefficients are rational, reduce to handling singularities in the
# same half-plane as the base point.
need_conjugates = _try_merge_conjugate_singularities(dop, sing, base, todo)
Scalars = ComplexBallField(utilities.prec_from_eps(eps))
id_mat = matrix.identity_matrix(Scalars, dop.order())
def matprod(elts):
return prod(reversed(elts), id_mat)
for key, point in list(todo.items()):
# We could call _local_monodromy_loop() if point is irregular, but
# delaying it may allow us to start returning results earlier.
if point.is_regular():
mon, scalar = _formal_monodromy_naive(dop.shift(point), Scalars)
if scalar:
# No need to compute the connection matrices then!
# XXX When we do need them, though, it would be better to get
# the formal monodromy as a byproduct of their computation.
if point.want_self:
yield LocalMonodromyData(QQbar(point), mon, True)
if point.want_conj:
conj = point.conjugate()
logger.info(
"Computing local monodromy around %s by "
"complex conjugation", conj)
conj_mat = ~mon.conjugate()
yield LocalMonodromyData(QQbar(conj), conj_mat, True)
if point is not base:
del todo[key]
continue
point.local_monodromy = [mon]
point.polygon = [point]
if need_conjugates:
base_conj_mat = dop.numerical_transition_matrix(
[base, base.conjugate()], eps, assume_analytic=True)
def conjugate_monodromy(mat):
return ~base_conj_mat * ~mat.conjugate() * base_conj_mat
tree = _spanning_tree(base, todo.values())
def dfs(x, path, path_mat):
logger.info("Computing local monodromy around %s via %s", x, path)
local_mat = matprod(x.local_monodromy)
based_mat = (~path_mat) * local_mat * path_mat
if x.want_self:
yield LocalMonodromyData(QQbar(x), based_mat, False)
if x.want_conj:
conj = x.conjugate()
logger.info(
"Computing local monodromy around %s by complex "
"conjugation", conj)
conj_mat = conjugate_monodromy(based_mat)
yield LocalMonodromyData(QQbar(x), conj_mat, False)
x.done = True
for y in tree.neighbors(x):
if y.done:
continue
if y.local_monodromy is None:
y.polygon, y.local_monodromy = _local_monodromy_loop(
dop, y, eps)
new_path_mat = _extend_path_mat(dop, path_mat, x, y, eps, matprod)
yield from dfs(y, path + [y], new_path_mat)
yield from dfs(base, [base], id_mat)
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
def conjugate(self):
value = QQbar.coerce(self.value).conjugate()
return Point(value, self.dop, **self.options)