def __init__(self, X):
"""
Create a Set_object
This function is called by the Set function; users
shouldn't call this directly.
EXAMPLES::
sage: type(Set(QQ))
<class 'sage.sets.set.Set_object_with_category'>
TESTS::
sage: _a, _b = get_coercion_model().canonical_coercion(Set([0]), 0)
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents:
'<class 'sage.sets.set.Set_object_enumerated_with_category'>'
and 'Integer Ring'
"""
from sage.rings.integer import is_Integer
if isinstance(X, (int, long)) or is_Integer(X):
# The coercion model will try to call Set_object(0)
raise ValueError('underlying object cannot be an integer')
Parent.__init__(self, category=Sets())
self.__object = X
def __classcall__(cls, *args, **kwds):
"""
Normalize input to ensure a unique representation.
EXAMPLES::
sage: A = AffineSpace(2, GF(4,'a'))
sage: AffineGroup(A) is AffineGroup(2,4)
True
sage: AffineGroup(A) is AffineGroup(2, GF(4,'a'))
True
sage: A = AffineGroup(2, QQ)
sage: V = QQ^2
sage: A is AffineGroup(V)
True
"""
if len(args) == 1:
V = args[0]
if isinstance(V, AffineGroup):
return V
try:
degree = V.dimension_relative()
except AttributeError:
degree = V.dimension()
ring = V.base_ring()
if len(args) == 2:
degree, ring = args
from sage.rings.integer import is_Integer
if is_Integer(ring):
from sage.rings.finite_rings.finite_field_constructor import FiniteField
var = kwds.get('var', 'a')
ring = FiniteField(ring, var)
return super(AffineGroup, cls).__classcall__(cls, degree, ring)
def pyobject(self, ex, obj):
from mathics.core import expression
from mathics.core.expression import Number
if obj is None:
return expression.Symbol('Null')
elif isinstance(obj, (list, tuple)) or is_Vector(obj):
return expression.Expression('List', *(from_sage(item, self.subs) for item in obj))
elif isinstance(obj, Constant):
return expression.Symbol(obj._conversions.get('mathematica', obj._name))
elif is_Integer(obj):
return expression.Integer(str(obj))
elif isinstance(obj, sage.Rational):
rational = expression.Rational(str(obj))
if rational.value.denom() == 1:
return expression.Integer(rational.value.numer())
else:
return rational
elif isinstance(obj, sage.RealDoubleElement) or is_RealNumber(obj):
return expression.Real(str(obj))
elif is_ComplexNumber(obj):
real = Number.from_string(str(obj.real())).value
imag = Number.from_string(str(obj.imag())).value
return expression.Complex(real, imag)
elif isinstance(obj, NumberFieldElement_quadratic):
# TODO: this need not be a complex number, but we assume so!
real = Number.from_string(str(obj.real())).value
imag = Number.from_string(str(obj.imag())).value
return expression.Complex(real, imag)
else:
return expression.from_python(obj)
def __init__(self, degree, base_ring, category=None):
"""
Base class for matrix groups over generic base rings
You should not use this class directly. Instead, use one of
the more specialized derived classes.
INPUT:
- ``degree`` -- integer. The degree (matrix size) of the
matrix group.
- ``base_ring`` -- ring. The base ring of the matrices.
TESTS::
sage: G = GL(2, QQ)
sage: from sage.groups.matrix_gps.matrix_group import MatrixGroup_generic
sage: isinstance(G, MatrixGroup_generic)
True
"""
assert is_Ring(base_ring)
assert is_Integer(degree)
self._deg = degree
if self._deg <= 0:
raise ValueError('the degree must be at least 1')
if (category is None) and is_FiniteField(base_ring):
from sage.categories.finite_groups import FiniteGroups
category = FiniteGroups()
super(MatrixGroup_generic, self).__init__(base=base_ring, category=category)
def __init__(self, X):
"""
Create a Set_object
This function is called by the Set function; users
shouldn't call this directly.
EXAMPLES::
sage: type(Set(QQ))
<class 'sage.sets.set.Set_object_with_category'>
TESTS::
sage: _a, _b = get_coercion_model().canonical_coercion(Set([0]), 0)
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents:
'<class 'sage.sets.set.Set_object_enumerated_with_category'>'
and 'Integer Ring'
"""
from sage.rings.integer import is_Integer
if isinstance(X, (int,long)) or is_Integer(X):
# The coercion model will try to call Set_object(0)
raise ValueError('underlying object cannot be an integer')
Parent.__init__(self, category=Sets())
self.__object = X
def __classcall__(cls, *args, **kwds):
"""
Normalize input to ensure a unique representation.
EXAMPLES::
sage: A = AffineSpace(2, GF(4,'a'))
sage: AffineGroup(A) is AffineGroup(2,4)
True
sage: AffineGroup(A) is AffineGroup(2, GF(4,'a'))
True
sage: A = AffineGroup(2, QQ)
sage: V = QQ^2
sage: A is AffineGroup(V)
True
"""
if len(args) == 1:
V = args[0]
if isinstance(V, AffineGroup):
return V
try:
degree = V.dimension_relative()
except AttributeError:
degree = V.dimension()
ring = V.base_ring()
if len(args) == 2:
degree, ring = args
from sage.rings.integer import is_Integer
if is_Integer(ring):
from sage.rings.finite_rings.finite_field_constructor import FiniteField
var = kwds.get('var', 'a')
ring = FiniteField(ring, var)
return super(AffineGroup, cls).__classcall__(cls, degree, ring)
def _check_trac_number(trac_number):
"""
Check that the argument is likely to be a valid trac issue number.
INPUT:
- ``trac_number`` -- anything.
OUTPUT:
This function returns nothing. A ``ValueError`` is raised if the
argument can not be a valid trac number.
EXAMPLES::
sage: from sage.misc.superseded import _check_trac_number
sage: _check_trac_number(1)
sage: _check_trac_number(int(10))
sage: _check_trac_number(long(1000))
sage: _check_trac_number('1')
Traceback (most recent call last):
...
ValueError: The argument "1" is not a valid trac issue number.
"""
from sage.rings.integer import is_Integer
err = ValueError('The argument "'+str(trac_number)+'" is not a valid trac issue number.')
if not (is_Integer(trac_number) or isinstance(trac_number, (int,long))):
raise err
if trac_number < 0:
raise err
def _check_trac_number(trac_number):
"""
Check that the argument is likely to be a valid trac issue number.
INPUT:
- ``trac_number`` -- anything.
OUTPUT:
This function returns nothing. A ``ValueError`` is raised if the
argument can not be a valid trac number.
EXAMPLES::
sage: from sage.misc.superseded import _check_trac_number
sage: _check_trac_number(1)
sage: _check_trac_number(int(10))
sage: _check_trac_number(long(1000))
sage: _check_trac_number('1')
Traceback (most recent call last):
...
ValueError: The argument "1" is not a valid trac issue number.
"""
from sage.rings.integer import is_Integer
err = ValueError('The argument "' + str(trac_number) +
'" is not a valid trac issue number.')
if not (is_Integer(trac_number) or isinstance(trac_number, (int, long))):
raise err
if trac_number < 0:
raise err
def Polyhedron(vertices=None,
rays=None,
lines=None,
ieqs=None,
eqns=None,
ambient_dim=None,
base_ring=None,
minimize=True,
verbose=False,
backend=None):
"""
Construct a polyhedron object.
You may either define it with vertex/ray/line or
inequalities/equations data, but not both. Redundant data will
automatically be removed (unless ``minimize=False``), and the
complementary representation will be computed.
INPUT:
- ``vertices`` -- list of point. Each point can be specified as
any iterable container of ``base_ring`` elements. If ``rays`` or
``lines`` are specified but no ``vertices``, the origin is
taken to be the single vertex.
- ``rays`` -- list of rays. Each ray can be specified as any
iterable container of ``base_ring`` elements.
- ``lines`` -- list of lines. Each line can be specified as any
iterable container of ``base_ring`` elements.
- ``ieqs`` -- list of inequalities. Each line can be specified as any
iterable container of ``base_ring`` elements. An entry equal to
``[-1,7,3,4]`` represents the inequality `7x_1+3x_2+4x_3\geq 1`.
- ``eqns`` -- list of equalities. Each line can be specified as
any iterable container of ``base_ring`` elements. An entry equal to
``[-1,7,3,4]`` represents the equality `7x_1+3x_2+4x_3= 1`.
- ``base_ring`` -- a sub-field of the reals implemented in
Sage. The field over which the polyhedron will be defined. For
``QQ`` and algebraic extensions, exact arithmetic will be
used. For ``RDF``, floating point numbers will be used. Floating
point arithmetic is faster but might give the wrong result for
degenerate input.
- ``ambient_dim`` -- integer. The ambient space dimension. Usually
can be figured out automatically from the H/Vrepresentation
dimensions.
- ``backend`` -- string or ``None`` (default). The backend to use. Valid choices are
* ``'cdd'``: use cdd
(:mod:`~sage.geometry.polyhedron.backend_cdd`) with `\QQ` or
`\RDF` coefficients depending on ``base_ring``.
* ``'ppl'``: use ppl
(:mod:`~sage.geometry.polyhedron.backend_ppl`) with `\ZZ` or
`\QQ` coefficients depending on ``base_ring``.
* ``'field'``: use python implementation
(:mod:`~sage.geometry.polyhedron.backend_field`) for any field
Some backends support further optional arguments:
- ``minimize`` -- boolean (default: ``True``). Whether to
immediately remove redundant H/V-representation data. Currently
not used.
- ``verbose`` -- boolean (default: ``False``). Whether to print
verbose output for debugging purposes. Only supported by the cdd
backends.
OUTPUT:
The polyhedron defined by the input data.
EXAMPLES:
Construct some polyhedra::
sage: square_from_vertices = Polyhedron(vertices = [[1, 1], [1, -1], [-1, 1], [-1, -1]])
sage: square_from_ieqs = Polyhedron(ieqs = [[1, 0, 1], [1, 1, 0], [1, 0, -1], [1, -1, 0]])
sage: list(square_from_ieqs.vertex_generator())
[A vertex at (1, -1),
A vertex at (1, 1),
A vertex at (-1, 1),
A vertex at (-1, -1)]
sage: list(square_from_vertices.inequality_generator())
[An inequality (1, 0) x + 1 >= 0,
An inequality (0, 1) x + 1 >= 0,
An inequality (-1, 0) x + 1 >= 0,
An inequality (0, -1) x + 1 >= 0]
sage: p = Polyhedron(vertices = [[1.1, 2.2], [3.3, 4.4]], base_ring=RDF)
sage: p.n_inequalities()
2
The same polyhedron given in two ways::
sage: p = Polyhedron(ieqs = [[0,1,0,0],[0,0,1,0]])
sage: p.Vrepresentation()
(A line in the direction (0, 0, 1),
A ray in the direction (1, 0, 0),
A ray in the direction (0, 1, 0),
A vertex at (0, 0, 0))
sage: q = Polyhedron(vertices=[[0,0,0]], rays=[[1,0,0],[0,1,0]], lines=[[0,0,1]])
sage: q.Hrepresentation()
(An inequality (1, 0, 0) x + 0 >= 0,
An inequality (0, 1, 0) x + 0 >= 0)
Finally, a more complicated example. Take `\mathbb{R}_{\geq 0}^6` with
coordinates `a, b, \dots, f` and
* The inequality `e+b \geq c+d`
* The inequality `e+c \geq b+d`
* The equation `a+b+c+d+e+f = 31`
::
sage: positive_coords = Polyhedron(ieqs=[
... [0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0],
... [0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1]])
sage: P = Polyhedron(ieqs=positive_coords.inequalities() + (
... [0,0,1,-1,-1,1,0], [0,0,-1,1,-1,1,0]), eqns=[[-31,1,1,1,1,1,1]])
sage: P
A 5-dimensional polyhedron in QQ^6 defined as the convex hull of 7 vertices
sage: P.dim()
5
sage: P.Vrepresentation()
(A vertex at (31, 0, 0, 0, 0, 0), A vertex at (0, 0, 0, 0, 0, 31),
A vertex at (0, 0, 0, 0, 31, 0), A vertex at (0, 0, 31/2, 0, 31/2, 0),
A vertex at (0, 31/2, 31/2, 0, 0, 0), A vertex at (0, 31/2, 0, 0, 31/2, 0),
A vertex at (0, 0, 0, 31/2, 31/2, 0))
.. NOTE::
* Once constructed, a ``Polyhedron`` object is immutable.
* Although the option ``field=RDF`` allows numerical data to
be used, it might not give the right answer for degenerate
input data - the results can depend upon the tolerance
setting of cdd.
"""
# Clean up the arguments
vertices = _make_listlist(vertices)
rays = _make_listlist(rays)
lines = _make_listlist(lines)
ieqs = _make_listlist(ieqs)
eqns = _make_listlist(eqns)
got_Vrep = (len(vertices + rays + lines) > 0)
got_Hrep = (len(ieqs + eqns) > 0)
if got_Vrep and got_Hrep:
raise ValueError('You cannot specify both H- and V-representation.')
elif got_Vrep:
deduced_ambient_dim = _common_length_of(vertices, rays, lines)[1]
elif got_Hrep:
deduced_ambient_dim = _common_length_of(ieqs, eqns)[1] - 1
else:
if ambient_dim is None:
deduced_ambient_dim = 0
else:
deduced_ambient_dim = ambient_dim
if base_ring is None:
base_ring = ZZ
# set ambient_dim
if ambient_dim is not None and deduced_ambient_dim != ambient_dim:
raise ValueError(
'Ambient space dimension mismatch. Try removing the "ambient_dim" parameter.'
)
ambient_dim = deduced_ambient_dim
# figure out base_ring
from sage.misc.flatten import flatten
values = flatten(vertices + rays + lines + ieqs + eqns)
if base_ring is not None:
try:
convert = not all(x.parent() is base_ring for x in values)
except AttributeError: # No x.parent() method?
convert = True
else:
from sage.rings.integer import is_Integer
from sage.rings.rational import is_Rational
from sage.rings.real_double import is_RealDoubleElement
if all(is_Integer(x) for x in values):
if got_Vrep:
base_ring = ZZ
else: # integral inequalities usually do not determine a lattice polytope!
base_ring = QQ
convert = False
elif all(is_Rational(x) for x in values):
base_ring = QQ
convert = False
elif all(is_RealDoubleElement(x) for x in values):
base_ring = RDF
convert = False
else:
try:
for v in values:
ZZ(v)
if got_Vrep:
base_ring = ZZ
else:
base_ring = QQ
convert = True
except (TypeError, ValueError):
from sage.structure.sequence import Sequence
values = Sequence(values)
common_ring = values.universe()
if QQ.has_coerce_map_from(common_ring):
base_ring = QQ
convert = True
elif common_ring is RR: # DWIM: replace with RDF
base_ring = RDF
convert = True
else:
base_ring = common_ring
convert = True
# Add the origin if necesarry
if got_Vrep and len(vertices) == 0:
vertices = [[0] * ambient_dim]
# Specific backends can override the base_ring
from sage.geometry.polyhedron.parent import Polyhedra
parent = Polyhedra(base_ring, ambient_dim, backend=backend)
base_ring = parent.base_ring()
# finally, construct the Polyhedron
Hrep = Vrep = None
if got_Hrep:
Hrep = [ieqs, eqns]
if got_Vrep:
Vrep = [vertices, rays, lines]
return parent(Vrep, Hrep, convert=convert, verbose=verbose)
def __call__(self, P):
r"""
Returns a rational point P in the abstract Homset J(K), given:
0. A point P in J = Jac(C), returning P; 1. A point P on the curve
C such that J = Jac(C), where C is an odd degree model, returning
[P - oo]; 2. A pair of points (P, Q) on the curve C such that J =
Jac(C), returning [P-Q]; 2. A list of polynomials (a,b) such that
`b^2 + h*b - f = 0 mod a`, returning [(a(x),y-b(x))].
EXAMPLES::
sage: P.<x> = PolynomialRing(QQ)
sage: f = x^5 - x + 1; h = x
sage: C = HyperellipticCurve(f,h,'u,v')
sage: P = C(0,1,1)
sage: J = C.jacobian()
sage: Q = J(QQ)(P)
sage: for i in range(6): i*Q
(1)
(u, v - 1)
(u^2, v + u - 1)
(u^2, v + 1)
(u, v + 1)
(1)
::
sage: F.<a> = GF(3)
sage: R.<x> = F[]
sage: f = x^5-1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: X = J(F)
sage: a = x^2-x+1
sage: b = -x +1
sage: c = x-1
sage: d = 0
sage: D1 = X([a,b])
sage: D1
(x^2 + 2*x + 1, y + x + 2)
sage: D2 = X([c,d])
sage: D2
(x + 2, y)
sage: D1+D2
(x^2 + 2*x + 2, y + 2*x + 1)
"""
if isinstance(P, (int, long, Integer)) and P == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1), R(0)))
elif isinstance(P, (list, tuple)):
if len(P) == 1 and P[0] == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1), R(0)))
elif len(P) == 2:
P1 = P[0]
P2 = P[1]
if is_Integer(P1) and is_Integer(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
P2 = R(P2)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Integer(P1) and is_Polynomial(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Integer(P2) and is_Polynomial(P1):
R = PolynomialRing(self.value_ring(), 'x')
P2 = R(P2)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Polynomial(P1) and is_Polynomial(P2):
return JacobianMorphism_divisor_class_field(self, tuple(P))
if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
return self(P1) - self(P2)
raise TypeError("Argument P (= %s) must have length 2." % P)
elif isinstance(
P,
JacobianMorphism_divisor_class_field) and self == P.parent():
return P
elif is_SchemeMorphism(P):
x0 = P[0]
y0 = P[1]
R, x = PolynomialRing(self.value_ring(), 'x').objgen()
return self((x - x0, R(y0)))
raise TypeError(
"Argument P (= %s) does not determine a divisor class" % P)
def to_cartesian(self, func, params=None):
"""
Returns a 3-tuple of functions, parameterized over ``params``, that
represents the Cartesian coordinates of the value of ``func``.
INPUT:
- ``func`` - A function in this coordinate space. Corresponds to the
independent variable.
- ``params`` - The parameters of ``func``. Corresponds to the dependent
variables.
EXAMPLES::
sage: from sage.plot.plot3d.plot3d import _ArbitraryCoordinates
sage: x, y, z = var('x y z')
sage: T = _ArbitraryCoordinates((x + y, x - y, z), z,[x,y])
sage: f(x, y) = 2*x+y
sage: T.to_cartesian(f, [x, y])
(x + y, x - y, 2*x + y)
sage: [h(1,2) for h in T.to_cartesian(lambda x,y: 2*x+y)]
[3.0, -1.0, 4.0]
We try to return a function having the same variable names as
the function passed in::
sage: from sage.plot.plot3d.plot3d import _ArbitraryCoordinates
sage: x, y, z = var('x y z')
sage: T = _ArbitraryCoordinates((x + y, x - y, z), z,[x,y])
sage: f(a, b) = 2*a+b
sage: T.to_cartesian(f, [a, b])
(a + b, a - b, 2*a + b)
sage: t1,t2,t3=T.to_cartesian(lambda a,b: 2*a+b)
sage: import inspect
sage: inspect.getargspec(t1)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: inspect.getargspec(t2)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: inspect.getargspec(t3)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: def g(a,b): return 2*a+b
sage: t1,t2,t3=T.to_cartesian(g)
sage: inspect.getargspec(t1)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: t1,t2,t3=T.to_cartesian(2*a+b)
sage: inspect.getargspec(t1)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
If we cannot guess the right parameter names, then the
parameters are named `u` and `v`::
sage: from sage.plot.plot3d.plot3d import _ArbitraryCoordinates
sage: x, y, z = var('x y z')
sage: T = _ArbitraryCoordinates((x + y, x - y, z), z,[x,y])
sage: t1,t2,t3=T.to_cartesian(operator.add)
sage: inspect.getargspec(t1)
ArgSpec(args=['u', 'v'], varargs=None, keywords=None, defaults=None)
sage: [h(1,2) for h in T.to_cartesian(operator.mul)]
[3.0, -1.0, 2.0]
sage: [h(u=1,v=2) for h in T.to_cartesian(operator.mul)]
[3.0, -1.0, 2.0]
The output of the function ``func`` is coerced to a float when
it is evaluated if the function is something like a lambda or
python callable. This takes care of situations like f returning a
singleton numpy array, for example.
sage: from numpy import array
sage: v_phi=array([ 0., 1.57079637, 3.14159274, 4.71238911, 6.28318548])
sage: v_theta=array([ 0., 0.78539819, 1.57079637, 2.35619456, 3.14159274])
sage: m_r=array([[ 0.16763356, 0.25683223, 0.16649297, 0.10594339, 0.55282422],
....: [ 0.16763356, 0.19993708, 0.31403568, 0.47359696, 0.55282422],
....: [ 0.16763356, 0.25683223, 0.16649297, 0.10594339, 0.55282422],
....: [ 0.16763356, 0.19993708, 0.31403568, 0.47359696, 0.55282422],
....: [ 0.16763356, 0.25683223, 0.16649297, 0.10594339, 0.55282422]])
sage: import scipy.interpolate
sage: f=scipy.interpolate.RectBivariateSpline(v_phi,v_theta,m_r)
sage: spherical_plot3d(f,(0,2*pi),(0,pi))
Graphics3d Object
"""
from sage.symbolic.expression import is_Expression
from sage.rings.real_mpfr import is_RealNumber
from sage.rings.integer import is_Integer
if params is not None and (is_Expression(func) or is_RealNumber(func)
or is_Integer(func)):
return self.transform(
**{
self.dep_var: func,
self.indep_vars[0]: params[0],
self.indep_vars[1]: params[1]
})
else:
# func might be a lambda or a Python callable; this makes it slightly
# more complex.
import sage.symbolic.ring
dep_var_dummy = sage.symbolic.ring.var(self.dep_var)
indep_var_dummies = sage.symbolic.ring.var(','.join(
self.indep_vars))
transformation = self.transform(
**{
self.dep_var: dep_var_dummy,
self.indep_vars[0]: indep_var_dummies[0],
self.indep_vars[1]: indep_var_dummies[1]
})
if params is None:
if callable(func):
params = _find_arguments_for_callable(func)
if params is None:
params = ['u', 'v']
else:
raise ValueError("function is not callable")
def subs_func(t):
# We use eval so that the lambda function has the same
# variable names as the original function
ll = """lambda {x},{y}: t.subs({{
dep_var_dummy: float(func({x}, {y})),
indep_var_dummies[0]: float({x}),
indep_var_dummies[1]: float({y})
}})""".format(x=params[0], y=params[1])
return eval(
ll,
dict(t=t,
func=func,
dep_var_dummy=dep_var_dummy,
indep_var_dummies=indep_var_dummies))
return [subs_func(_) for _ in transformation]
def __call__(self, P):
r"""
Returns a rational point P in the abstract Homset J(K), given:
0. A point P in J = Jac(C), returning P; 1. A point P on the curve
C such that J = Jac(C), where C is an odd degree model, returning
[P - oo]; 2. A pair of points (P, Q) on the curve C such that J =
Jac(C), returning [P-Q]; 2. A list of polynomials (a,b) such that
`b^2 + h*b - f = 0 mod a`, returning [(a(x),y-b(x))].
EXAMPLES::
sage: P.<x> = PolynomialRing(QQ)
sage: f = x^5 - x + 1; h = x
sage: C = HyperellipticCurve(f,h,'u,v')
sage: P = C(0,1,1)
sage: J = C.jacobian()
sage: Q = J(QQ)(P)
sage: for i in range(6): i*Q
(1)
(u, v - 1)
(u^2, v + u - 1)
(u^2, v + 1)
(u, v + 1)
(1)
::
sage: F.<a> = GF(3)
sage: R.<x> = F[]
sage: f = x^5-1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: X = J(F)
sage: a = x^2-x+1
sage: b = -x +1
sage: c = x-1
sage: d = 0
sage: D1 = X([a,b])
sage: D1
(x^2 + 2*x + 1, y + x + 2)
sage: D2 = X([c,d])
sage: D2
(x + 2, y)
sage: D1+D2
(x^2 + 2*x + 2, y + 2*x + 1)
"""
if isinstance(P,(int,long,Integer)) and P == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1),R(0)))
elif isinstance(P,(list,tuple)):
if len(P) == 1 and P[0] == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1),R(0)))
elif len(P) == 2:
P1 = P[0]
P2 = P[1]
if is_Integer(P1) and is_Integer(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
P2 = R(P2)
return JacobianMorphism_divisor_class_field(self, tuple([P1,P2]))
if is_Integer(P1) and is_Polynomial(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
return JacobianMorphism_divisor_class_field(self, tuple([P1,P2]))
if is_Integer(P2) and is_Polynomial(P1):
R = PolynomialRing(self.value_ring(), 'x')
P2 = R(P2)
return JacobianMorphism_divisor_class_field(self, tuple([P1,P2]))
if is_Polynomial(P1) and is_Polynomial(P2):
return JacobianMorphism_divisor_class_field(self, tuple(P))
if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
return self(P1) - self(P2)
raise TypeError("Argument P (= %s) must have length 2."%P)
elif isinstance(P,JacobianMorphism_divisor_class_field) and self == P.parent():
return P
elif is_SchemeMorphism(P):
x0 = P[0]; y0 = P[1]
R, x = PolynomialRing(self.value_ring(), 'x').objgen()
return self((x-x0,R(y0)))
raise TypeError("Argument P (= %s) does not determine a divisor class"%P)
def to_cartesian(self, func, params=None):
"""
Returns a 3-tuple of functions, parameterized over ``params``, that
represents the Cartesian coordinates of the value of ``func``.
INPUT:
- ``func`` - A function in this coordinate space. Corresponds to the
independent variable.
- ``params`` - The parameters of func. Corresponds to the dependent
variables.
EXAMPLE::
sage: from sage.plot.plot3d.plot3d import _ArbitraryCoordinates
sage: x, y, z = var('x y z')
sage: T = _ArbitraryCoordinates((x + y, x - y, z), z,[x,y])
sage: f(x, y) = 2*x+y
sage: T.to_cartesian(f, [x, y])
(x + y, x - y, 2*x + y)
sage: [h(1,2) for h in T.to_cartesian(lambda x,y: 2*x+y)]
[3.0, -1.0, 4.0]
We try to return a function having the same variable names as
the function passed in::
sage: from sage.plot.plot3d.plot3d import _ArbitraryCoordinates
sage: x, y, z = var('x y z')
sage: T = _ArbitraryCoordinates((x + y, x - y, z), z,[x,y])
sage: f(a, b) = 2*a+b
sage: T.to_cartesian(f, [a, b])
(a + b, a - b, 2*a + b)
sage: t1,t2,t3=T.to_cartesian(lambda a,b: 2*a+b)
sage: import inspect
sage: inspect.getargspec(t1)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: inspect.getargspec(t2)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: inspect.getargspec(t3)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: def g(a,b): return 2*a+b
sage: t1,t2,t3=T.to_cartesian(g)
sage: inspect.getargspec(t1)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: t1,t2,t3=T.to_cartesian(2*a+b)
sage: inspect.getargspec(t1)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
If we cannot guess the right parameter names, then the
parameters are named `u` and `v`::
sage: from sage.plot.plot3d.plot3d import _ArbitraryCoordinates
sage: x, y, z = var('x y z')
sage: T = _ArbitraryCoordinates((x + y, x - y, z), z,[x,y])
sage: t1,t2,t3=T.to_cartesian(operator.add)
sage: inspect.getargspec(t1)
ArgSpec(args=['u', 'v'], varargs=None, keywords=None, defaults=None)
sage: [h(1,2) for h in T.to_cartesian(operator.mul)]
[3.0, -1.0, 2.0]
sage: [h(u=1,v=2) for h in T.to_cartesian(operator.mul)]
[3.0, -1.0, 2.0]
The output of the function `func` is coerced to a float when
it is evaluated if the function is something like a lambda or
python callable. This takes care of situations like f returning a
singleton numpy array, for example.
sage: from numpy import array
sage: v_phi=array([ 0., 1.57079637, 3.14159274, 4.71238911, 6.28318548])
sage: v_theta=array([ 0., 0.78539819, 1.57079637, 2.35619456, 3.14159274])
sage: m_r=array([[ 0.16763356, 0.25683223, 0.16649297, 0.10594339, 0.55282422],
... [ 0.16763356, 0.19993708, 0.31403568, 0.47359696, 0.55282422],
... [ 0.16763356, 0.25683223, 0.16649297, 0.10594339, 0.55282422],
... [ 0.16763356, 0.19993708, 0.31403568, 0.47359696, 0.55282422],
... [ 0.16763356, 0.25683223, 0.16649297, 0.10594339, 0.55282422]])
sage: import scipy.interpolate
sage: f=scipy.interpolate.RectBivariateSpline(v_phi,v_theta,m_r)
sage: spherical_plot3d(f,(0,2*pi),(0,pi))
Graphics3d Object
"""
from sage.symbolic.expression import is_Expression
from sage.rings.real_mpfr import is_RealNumber
from sage.rings.integer import is_Integer
if params is not None and (is_Expression(func) or is_RealNumber(func) or is_Integer(func)):
return self.transform(**{
self.dep_var: func,
self.indep_vars[0]: params[0],
self.indep_vars[1]: params[1]
})
else:
# func might be a lambda or a Python callable; this makes it slightly
# more complex.
import sage.symbolic.ring
dep_var_dummy = sage.symbolic.ring.var(self.dep_var)
indep_var_dummies = sage.symbolic.ring.var(','.join(self.indep_vars))
transformation = self.transform(**{
self.dep_var: dep_var_dummy,
self.indep_vars[0]: indep_var_dummies[0],
self.indep_vars[1]: indep_var_dummies[1]
})
if params is None:
if callable(func):
params = _find_arguments_for_callable(func)
if params is None:
params=['u','v']
else:
raise ValueError("function is not callable")
def subs_func(t):
# We use eval so that the lambda function has the same
# variable names as the original function
ll="""lambda {x},{y}: t.subs({{
dep_var_dummy: float(func({x}, {y})),
indep_var_dummies[0]: float({x}),
indep_var_dummies[1]: float({y})
}})""".format(x=params[0], y=params[1])
return eval(ll,dict(t=t, func=func, dep_var_dummy=dep_var_dummy,
indep_var_dummies=indep_var_dummies))
return [subs_func(_) for _ in transformation]
def Polyhedron(vertices=None, rays=None, lines=None,
ieqs=None, eqns=None,
ambient_dim=None, base_ring=None, minimize=True, verbose=False,
backend=None):
"""
Construct a polyhedron object.
You may either define it with vertex/ray/line or
inequalities/equations data, but not both. Redundant data will
automatically be removed (unless ``minimize=False``), and the
complementary representation will be computed.
INPUT:
- ``vertices`` -- list of point. Each point can be specified as
any iterable container of ``base_ring`` elements. If ``rays`` or
``lines`` are specified but no ``vertices``, the origin is
taken to be the single vertex.
- ``rays`` -- list of rays. Each ray can be specified as any
iterable container of ``base_ring`` elements.
- ``lines`` -- list of lines. Each line can be specified as any
iterable container of ``base_ring`` elements.
- ``ieqs`` -- list of inequalities. Each line can be specified as any
iterable container of ``base_ring`` elements. An entry equal to
``[-1,7,3,4]`` represents the inequality `7x_1+3x_2+4x_3\geq 1`.
- ``eqns`` -- list of equalities. Each line can be specified as
any iterable container of ``base_ring`` elements. An entry equal to
``[-1,7,3,4]`` represents the equality `7x_1+3x_2+4x_3= 1`.
- ``base_ring`` -- either ``QQ`` or ``RDF``. The field over which
the polyhedron will be defined. For ``QQ``, exact arithmetic
will be used. For ``RDF``, floating point numbers will be
used. Floating point arithmetic is faster but might give the
wrong result for degenerate input.
- ``ambient_dim`` -- integer. The ambient space dimension. Usually
can be figured out automatically from the H/Vrepresentation
dimensions.
- ``backend`` -- string or ``None`` (default). The backend to use. Valid choices are
* ``'cdd'``: use cdd
(:mod:`~sage.geometry.polyhedron.backend_cdd`) with `\QQ` or
`\RDF` coefficients depending on ``base_ring``.
* ``'ppl'``: use ppl
(:mod:`~sage.geometry.polyhedron.backend_ppl`) with `\ZZ` or
`\QQ` coefficients depending on ``base_ring``.
Some backends support further optional arguments:
- ``minimize`` -- boolean (default: ``True``). Whether to
immediately remove redundant H/V-representation data. Currently
not used.
- ``verbose`` -- boolean (default: ``False``). Whether to print
verbose output for debugging purposes. Only supported by the cdd
backends.
OUTPUT:
The polyhedron defined by the input data.
EXAMPLES:
Construct some polyhedra::
sage: square_from_vertices = Polyhedron(vertices = [[1, 1], [1, -1], [-1, 1], [-1, -1]])
sage: square_from_ieqs = Polyhedron(ieqs = [[1, 0, 1], [1, 1, 0], [1, 0, -1], [1, -1, 0]])
sage: list(square_from_ieqs.vertex_generator())
[A vertex at (1, -1),
A vertex at (1, 1),
A vertex at (-1, 1),
A vertex at (-1, -1)]
sage: list(square_from_vertices.inequality_generator())
[An inequality (1, 0) x + 1 >= 0,
An inequality (0, 1) x + 1 >= 0,
An inequality (-1, 0) x + 1 >= 0,
An inequality (0, -1) x + 1 >= 0]
sage: p = Polyhedron(vertices = [[1.1, 2.2], [3.3, 4.4]], base_ring=RDF)
sage: p.n_inequalities()
2
The same polyhedron given in two ways::
sage: p = Polyhedron(ieqs = [[0,1,0,0],[0,0,1,0]])
sage: p.Vrepresentation()
(A line in the direction (0, 0, 1),
A ray in the direction (1, 0, 0),
A ray in the direction (0, 1, 0),
A vertex at (0, 0, 0))
sage: q = Polyhedron(vertices=[[0,0,0]], rays=[[1,0,0],[0,1,0]], lines=[[0,0,1]])
sage: q.Hrepresentation()
(An inequality (1, 0, 0) x + 0 >= 0,
An inequality (0, 1, 0) x + 0 >= 0)
Finally, a more complicated example. Take `\mathbb{R}_{\geq 0}^6` with
coordinates `a, b, \dots, f` and
* The inequality `e+b \geq c+d`
* The inequality `e+c \geq b+d`
* The equation `a+b+c+d+e+f = 31`
::
sage: positive_coords = Polyhedron(ieqs=[
... [0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0],
... [0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1]])
sage: P = Polyhedron(ieqs=positive_coords.inequalities() + (
... [0,0,1,-1,-1,1,0], [0,0,-1,1,-1,1,0]), eqns=[[-31,1,1,1,1,1,1]])
sage: P
A 5-dimensional polyhedron in QQ^6 defined as the convex hull of 7 vertices
sage: P.dim()
5
sage: P.Vrepresentation()
(A vertex at (31, 0, 0, 0, 0, 0), A vertex at (0, 0, 0, 0, 0, 31),
A vertex at (0, 0, 0, 0, 31, 0), A vertex at (0, 0, 31/2, 0, 31/2, 0),
A vertex at (0, 31/2, 31/2, 0, 0, 0), A vertex at (0, 31/2, 0, 0, 31/2, 0),
A vertex at (0, 0, 0, 31/2, 31/2, 0))
.. NOTE::
* Once constructed, a ``Polyhedron`` object is immutable.
* Although the option ``field=RDF`` allows numerical data to
be used, it might not give the right answer for degenerate
input data - the results can depend upon the tolerance
setting of cdd.
"""
# Clean up the arguments
vertices = _make_listlist(vertices)
rays = _make_listlist(rays)
lines = _make_listlist(lines)
ieqs = _make_listlist(ieqs)
eqns = _make_listlist(eqns)
got_Vrep = (len(vertices+rays+lines) > 0)
got_Hrep = (len(ieqs+eqns) > 0)
if got_Vrep and got_Hrep:
raise ValueError('You cannot specify both H- and V-representation.')
elif got_Vrep:
deduced_ambient_dim = _common_length_of(vertices, rays, lines)[1]
elif got_Hrep:
deduced_ambient_dim = _common_length_of(ieqs, eqns)[1] - 1
else:
if ambient_dim is None:
deduced_ambient_dim = 0
else:
deduced_ambient_dim = ambient_dim
if base_ring is None:
base_ring = ZZ
# set ambient_dim
if ambient_dim is not None and deduced_ambient_dim!=ambient_dim:
raise ValueError('Ambient space dimension mismatch. Try removing the "ambient_dim" parameter.')
ambient_dim = deduced_ambient_dim
# figure out base_ring
from sage.misc.flatten import flatten
values = flatten(vertices+rays+lines+ieqs+eqns)
if base_ring is not None:
try:
convert = not all(x.parent() is base_ring for x in values)
except AttributeError: # No x.parent() method?
convert = True
else:
from sage.rings.integer import is_Integer
from sage.rings.rational import is_Rational
from sage.rings.real_double import is_RealDoubleElement
if all(is_Integer(x) for x in values):
if got_Vrep:
base_ring = ZZ
else: # integral inequalities usually do not determine a latice polytope!
base_ring = QQ
convert=False
elif all(is_Rational(x) for x in values):
base_ring = QQ
convert=False
elif all(is_RealDoubleElement(x) for x in values):
base_ring = RDF
convert=False
else:
try:
map(ZZ, values)
if got_Vrep:
base_ring = ZZ
else:
base_ring = QQ
convert = True
except TypeError:
from sage.structure.sequence import Sequence
values = Sequence(values)
if QQ.has_coerce_map_from(values.universe()):
base_ring = QQ
convert = True
else:
base_ring = RDF
convert = True
# Add the origin if necesarry
if got_Vrep and len(vertices)==0:
vertices = [ [0]*ambient_dim ]
# Specific backends can override the base_ring
from sage.geometry.polyhedron.parent import Polyhedra
parent = Polyhedra(base_ring, ambient_dim, backend=backend)
base_ring = parent.base_ring()
# Convert into base_ring if necessary
def convert_base_ring(lstlst):
return [ [base_ring(x) for x in lst] for lst in lstlst]
Hrep = Vrep = None
if got_Hrep:
Hrep = [ieqs, eqns]
if got_Vrep:
Vrep = [vertices, rays, lines]
# finally, construct the Polyhedron
return parent(Vrep, Hrep, convert=convert)
def to_cartesian(self, func, params=None):
"""
Returns a 3-tuple of functions, parameterized over ``params``, that
represents the cartesian coordinates of the value of ``func``.
INPUT:
- ``func`` - A function in this coordinate space. Corresponds to the
independent variable.
- ``params`` - The parameters of func. Corresponds to the dependent
variables.
EXAMPLE::
sage: from sage.plot.plot3d.plot3d import _ArbitraryCoordinates
sage: x, y, z = var('x y z')
sage: T = _ArbitraryCoordinates((x + y, x - y, z), z,[x,y])
sage: f(x, y) = 2*x+y
sage: T.to_cartesian(f, [x, y])
(x + y, x - y, 2*x + y)
sage: [h(1,2) for h in T.to_cartesian(lambda x,y: 2*x+y)]
[3, -1, 4]
We try to return a function having the same variable names as
the function passed in::
sage: from sage.plot.plot3d.plot3d import _ArbitraryCoordinates
sage: x, y, z = var('x y z')
sage: T = _ArbitraryCoordinates((x + y, x - y, z), z,[x,y])
sage: f(a, b) = 2*a+b
sage: T.to_cartesian(f, [a, b])
(a + b, a - b, 2*a + b)
sage: t1,t2,t3=T.to_cartesian(lambda a,b: 2*a+b)
sage: import inspect
sage: inspect.getargspec(t1)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: inspect.getargspec(t2)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: inspect.getargspec(t3)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: def g(a,b): return 2*a+b
sage: t1,t2,t3=T.to_cartesian(g)
sage: inspect.getargspec(t1)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
sage: t1,t2,t3=T.to_cartesian(2*a+b)
sage: inspect.getargspec(t1)
ArgSpec(args=['a', 'b'], varargs=None, keywords=None, defaults=None)
If we cannot guess the right parameter names, then the
parameters are named `u` and `v`::
sage: from sage.plot.plot3d.plot3d import _ArbitraryCoordinates
sage: x, y, z = var('x y z')
sage: T = _ArbitraryCoordinates((x + y, x - y, z), z,[x,y])
sage: t1,t2,t3=T.to_cartesian(operator.add)
sage: inspect.getargspec(t1)
ArgSpec(args=['u', 'v'], varargs=None, keywords=None, defaults=None)
sage: [h(1,2) for h in T.to_cartesian(operator.mul)]
[3, -1, 2]
sage: [h(u=1,v=2) for h in T.to_cartesian(operator.mul)]
[3, -1, 2]
"""
from sage.symbolic.expression import is_Expression
from sage.rings.real_mpfr import is_RealNumber
from sage.rings.integer import is_Integer
if params is not None and (is_Expression(func) or is_RealNumber(func) or is_Integer(func)):
return self.transform(**{
self.dep_var: func,
self.indep_vars[0]: params[0],
self.indep_vars[1]: params[1]
})
else:
# func might be a lambda or a Python callable; this makes it slightly
# more complex.
import sage.symbolic.ring
dep_var_dummy = sage.symbolic.ring.var(self.dep_var)
indep_var_dummies = sage.symbolic.ring.var(','.join(self.indep_vars))
transformation = self.transform(**{
self.dep_var: dep_var_dummy,
self.indep_vars[0]: indep_var_dummies[0],
self.indep_vars[1]: indep_var_dummies[1]
})
if params is None:
if callable(func):
params = _find_arguments_for_callable(func)
if params is None:
params=['u','v']
else:
raise ValueError, "function is not callable"
def subs_func(t):
# We use eval so that the lambda function has the same
# variable names as the original function
ll="""lambda {x},{y}: t.subs({{
dep_var_dummy: func({x}, {y}),
indep_var_dummies[0]: {x},
indep_var_dummies[1]: {y}
}})""".format(x=params[0], y=params[1])
return eval(ll,dict(t=t, func=func, dep_var_dummy=dep_var_dummy,
indep_var_dummies=indep_var_dummies))
return map(subs_func, transformation)