def _element_constructor_(self, x):
r"""
Create an element in this group from ``x``.
INPUT:
- ``x`` -- a rational number
TESTS::
sage: sys.path.append(os.getcwd()); from mac_lane import * # optional: standalone
sage: DiscreteValueGroup(0)(0)
0
sage: DiscreteValueGroup(0)(1)
Traceback (most recent call last):
...
ValueError: `1` is not in Trivial Additive Abelian Group.
sage: DiscreteValueGroup(1)(1)
1
sage: DiscreteValueGroup(1)(1/2)
Traceback (most recent call last):
...
ValueError: `1/2` is not in Additive Abelian Group generated by 1.
"""
x = QQ.coerce(x)
if x == 0 or (self._generator != 0 and x / self._generator in ZZ):
return x
raise ValueError("`{0}` is not in {1}.".format(x, self))
def _element_constructor_(self, x):
r"""
Create an element from ``x``.
INPUT:
- ``x`` -- a rational number or `\infty`
TESTS::
sage: sys.path.append(os.getcwd()); from mac_lane import * # optional: standalone
sage: DiscreteValuationCodomain()(0)
0
sage: DiscreteValuationCodomain()(infinity)
+Infinity
sage: DiscreteValuationCodomain()(-infinity)
-Infinity
"""
if x is infinity:
return x
if x is -infinity:
return x
if x not in QQ:
raise ValueError("must be a rational number or infinity")
return QQ.coerce(x)
def _element_constructor_(self, x):
r"""
Create an element in this group from ``x``.
INPUT:
- ``x`` -- a rational number
TESTS::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: DiscreteValueSemigroup([])(0)
0
sage: DiscreteValueSemigroup([])(1)
Traceback (most recent call last):
...
ValueError: `1` is not in Trivial Additive Abelian Semigroup.
sage: DiscreteValueSemigroup([1])(1)
1
sage: DiscreteValueSemigroup([1])(-1)
Traceback (most recent call last):
...
ValueError: `-1` is not in Additive Abelian Semigroup generated by 1.
"""
x = QQ.coerce(x)
if x in self._generators or self._solve_linear_program(x) is not None:
return x
raise ValueError("`{0}` is not in {1}.".format(x,self))
def _element_constructor_(self, x):
r"""
Create an element in this group from ``x``.
INPUT:
- ``x`` -- a rational number
TESTS::
sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(0)(0)
0
sage: DiscreteValueGroup(0)(1)
Traceback (most recent call last):
...
ValueError: `1` is not in Trivial Additive Abelian Group.
sage: DiscreteValueGroup(1)(1)
1
sage: DiscreteValueGroup(1)(1/2)
Traceback (most recent call last):
...
ValueError: `1/2` is not in Additive Abelian Group generated by 1.
"""
x = QQ.coerce(x)
if x == 0 or (self._generator != 0 and x/self._generator in ZZ):
return x
raise ValueError("`{0}` is not in {1}.".format(x,self))
def _element_constructor_(self, x):
r"""
Create an element from ``x``.
INPUT:
- ``x`` -- a rational number or `\infty`
TESTS::
sage: from sage.rings.valuation.value_group import DiscreteValuationCodomain
sage: DiscreteValuationCodomain()(0)
0
sage: DiscreteValuationCodomain()(infinity)
+Infinity
sage: DiscreteValuationCodomain()(-infinity)
-Infinity
"""
if x is infinity:
return x
if x is -infinity:
return x
if x not in QQ:
raise ValueError("must be a rational number or infinity")
return QQ.coerce(x)
def _element_constructor_(self, x):
r"""
Create an element from ``x``.
INPUT:
- ``x`` -- a rational number or `\infty`
TESTS::
sage: from sage.rings.valuation.value_group import DiscreteValuationCodomain
sage: DiscreteValuationCodomain()(0)
0
sage: DiscreteValuationCodomain()(infinity)
+Infinity
sage: DiscreteValuationCodomain()(-infinity)
-Infinity
"""
if x is infinity:
return x
if x is -infinity:
return x
if x not in QQ:
raise ValueError("must be a rational number or infinity")
return QQ.coerce(x)
def _element_constructor_(self, x):
r"""
Create an element in this group from ``x``.
INPUT:
- ``x`` -- a rational number
TESTS::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: DiscreteValueSemigroup([])(0)
0
sage: DiscreteValueSemigroup([])(1)
Traceback (most recent call last):
...
ValueError: `1` is not in Trivial Additive Abelian Semigroup.
sage: DiscreteValueSemigroup([1])(1)
1
sage: DiscreteValueSemigroup([1])(-1)
Traceback (most recent call last):
...
ValueError: `-1` is not in Additive Abelian Semigroup generated by 1.
"""
x = QQ.coerce(x)
if x in self._generators or self._solve_linear_program(x) is not None:
return x
raise ValueError("`{0}` is not in {1}.".format(x,self))
def _element_constructor_(self, x):
r"""
Create an element in this group from ``x``.
INPUT:
- ``x`` -- a rational number
TESTS::
sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(0)(0)
0
sage: DiscreteValueGroup(0)(1)
Traceback (most recent call last):
...
ValueError: `1` is not in Trivial Additive Abelian Group.
sage: DiscreteValueGroup(1)(1)
1
sage: DiscreteValueGroup(1)(1/2)
Traceback (most recent call last):
...
ValueError: `1/2` is not in Additive Abelian Group generated by 1.
"""
x = QQ.coerce(x)
if x == 0 or (self._generator != 0 and x/self._generator in ZZ):
return x
raise ValueError("`{0}` is not in {1}.".format(x,self))
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 _element_constructor_(self, x):
r"""
Create an element in this group from ``x``.
INPUT:
- ``x`` -- a rational number
TESTS::
sage: sys.path.append(os.getcwd()); from mac_lane import * # optional: standalone
sage: DiscreteValueSemigroup([])(0)
0
sage: DiscreteValueSemigroup([])(1)
Traceback (most recent call last):
...
ValueError: `1` is not in Trivial Additive Abelian Semigroup.
sage: DiscreteValueSemigroup([1])(1)
1
sage: DiscreteValueSemigroup([1])(-1)
Traceback (most recent call last):
...
ValueError: `-1` is not in Additive Abelian Semigroup generated by 1.
"""
x = QQ.coerce(x)
if x in self._generators or self._solve_linear_program(x) is not None:
return x
raise ValueError("`{0}` is not in {1}.".format(x, self))
def create_key(self, base, s):
r"""
Create a key which uniquely identifies a valuation.
TESTS::
sage: 3*ZZ.valuation(2) is 2*(3/2*ZZ.valuation(2)) # indirect doctest
True
"""
from sage.rings.all import infinity, QQ
if s is infinity or s not in QQ or s <= 0:
# for these values we can not return a TrivialValuation() in
# create_object() because that would override that instance's
# _factory_data and lead to pickling errors
raise ValueError("s must be a positive rational")
if base.is_trivial():
# for the same reason we can not accept trivial valuations here
raise ValueError("base must not be trivial")
s = QQ.coerce(s)
if s == 1:
# we would override the _factory_data of base if we just returned
# it in create_object() so we just refuse to do so
raise ValueError("s must not be 1")
if isinstance(base, ScaledValuation_generic):
return self.create_key(base._base_valuation, s*base._scale)
return base, s
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 __classcall__(cls, generator):
r"""
Normalizes ``generator`` to a positive rational so that this is a
unique parent.
TESTS::
sage: sys.path.append(os.getcwd()); from mac_lane import * # optional: standalone
sage: DiscreteValueGroup(1) is DiscreteValueGroup(-1)
True
"""
generator = QQ.coerce(generator)
generator = generator.abs()
return super(DiscreteValueGroup, cls).__classcall__(cls, generator)
def __classcall__(cls, generator):
r"""
Normalizes ``generator`` to a positive rational so that this is a
unique parent.
TESTS::
sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(1) is DiscreteValueGroup(-1)
True
"""
generator = QQ.coerce(generator)
generator = generator.abs()
return super(DiscreteValueGroup, cls).__classcall__(cls, generator)
def __classcall__(cls, generator):
r"""
Normalizes ``generator`` to a positive rational so that this is a
unique parent.
TESTS::
sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(1) is DiscreteValueGroup(-1)
True
"""
generator = QQ.coerce(generator)
generator = generator.abs()
return super(DiscreteValueGroup, cls).__classcall__(cls, generator)
def some_elements(self):
r"""
Return some typical elements in this semigroup.
EXAMPLES::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: list(DiscreteValueSemigroup([-3/8,1/2]).some_elements())
[0, -3/8, 1/2, ...]
"""
yield self(0)
if self.is_trivial():
return
for g in self._generators:
yield g
from sage.rings.all import ZZ
for x in (ZZ**len(self._generators)).some_elements():
yield QQ.coerce(sum([abs(c)*g for c,g in zip(x,self._generators)]))
def some_elements(self):
r"""
Return some typical elements in this semigroup.
EXAMPLES::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: list(DiscreteValueSemigroup([-3/8,1/2]).some_elements())
[0, -3/8, 1/2, ...]
"""
yield self(0)
if self.is_trivial():
return
for g in self._generators:
yield g
from sage.rings.all import ZZ
for x in (ZZ**len(self._generators)).some_elements():
yield QQ.coerce(sum([abs(c)*g for c,g in zip(x,self._generators)]))
def some_elements(self):
r"""
Return some typical elements in this semigroup.
EXAMPLES::
sage: sys.path.append(os.getcwd()); from mac_lane import * # optional: standalone
sage: list(DiscreteValueSemigroup([-3/8,1/2]).some_elements())
[0, -3/8, 1/2, ...]
"""
yield self(0)
if self.is_trivial():
return
for g in self._generators:
yield g
from sage.rings.all import ZZ
for x in (ZZ**len(self._generators)).some_elements():
yield QQ.coerce(
sum([abs(c) * g for c, g in zip(x, self._generators)]))
def __classcall__(cls, generators):
r"""
Normalize ``generators``.
TESTS:
We do not find minimal generators or something like that but just sort the
generators and drop generators that are trivially contained in the
semigroup generated by the remaining generators::
sage: sys.path.append(os.getcwd()); from mac_lane import * # optional: standalone
sage: DiscreteValueSemigroup([1,2]) is DiscreteValueSemigroup([1])
True
In this case, the normalization is not sufficient to determine that
these are the same semigroup::
sage: DiscreteValueSemigroup([1,-1,1/3]) is DiscreteValueSemigroup([1/3,-1/3])
False
"""
if generators in QQ:
generators = [generators]
generators = list(set([QQ.coerce(g) for g in generators if g != 0]))
generators.sort()
simplified_generators = generators
# this is not very efficient but there should never be more than a
# couple of generators
for g in generators:
for h in generators:
if g == h: continue
from sage.rings.all import NN
if h / g in NN:
simplified_generators.remove(h)
break
return super(DiscreteValueSemigroup,
cls).__classcall__(cls, tuple(simplified_generators))
def __classcall__(cls, generators):
r"""
Normalize ``generators``.
TESTS:
We do not find minimal generators or something like that but just sort the
generators and drop generators that are trivially contained in the
semigroup generated by the remaining generators::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: DiscreteValueSemigroup([1,2]) is DiscreteValueSemigroup([1])
True
In this case, the normalization is not sufficient to determine that
these are the same semigroup::
sage: DiscreteValueSemigroup([1,-1,1/3]) is DiscreteValueSemigroup([1/3,-1/3])
False
"""
if generators in QQ:
generators = [generators]
generators = list(set([QQ.coerce(g) for g in generators if g != 0]))
generators.sort()
simplified_generators = generators
# this is not very efficient but there should never be more than a
# couple of generators
for g in generators:
for h in generators:
if g == h: continue
from sage.rings.all import NN
if h/g in NN:
simplified_generators.remove(h)
break
return super(DiscreteValueSemigroup, cls).__classcall__(cls, tuple(simplified_generators))
def _mul_(self, other, switch_sides=False):
r"""
Return the semigroup generated by ``other`` times the generators of this
semigroup.
INPUT:
- ``other`` -- a rational number
EXAMPLES::
sage: sys.path.append(os.getcwd()); from mac_lane import * # optional: standalone
sage: D = DiscreteValueSemigroup(1/2)
sage: 1/2 * D
Additive Abelian Semigroup generated by 1/4
sage: D * (1/2)
Additive Abelian Semigroup generated by 1/4
sage: D * 0
Trivial Additive Abelian Semigroup
"""
other = QQ.coerce(other)
return DiscreteValueSemigroup([g * other for g in self._generators])
def _mul_(self, other, switch_sides=False):
r"""
Return the semigroup generated by ``other`` times the generators of this
semigroup.
INPUT:
- ``other`` -- a rational number
EXAMPLES::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: D = DiscreteValueSemigroup(1/2)
sage: 1/2 * D
Additive Abelian Semigroup generated by 1/4
sage: D * (1/2)
Additive Abelian Semigroup generated by 1/4
sage: D * 0
Trivial Additive Abelian Semigroup
"""
other = QQ.coerce(other)
return DiscreteValueSemigroup([g*other for g in self._generators])
def _mul_(self, other, switch_sides=False):
r"""
Return the semigroup generated by ``other`` times the generators of this
semigroup.
INPUT:
- ``other`` -- a rational number
EXAMPLES::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: D = DiscreteValueSemigroup(1/2)
sage: 1/2 * D
Additive Abelian Semigroup generated by 1/4
sage: D * (1/2)
Additive Abelian Semigroup generated by 1/4
sage: D * 0
Trivial Additive Abelian Semigroup
"""
other = QQ.coerce(other)
return DiscreteValueSemigroup([g*other for g in self._generators])
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
