def sdp_design_matrix(self):
r"""
Return the incidence matrix of the SDP design of the bent function.
This method returns the incidence matrix of the design of type
:math:`R(\mathtt{self})`, as described by Dillon and Schatz [DS1987]_.
This is a design with the symmetric difference property [Kan1975]_.
INPUT:
- ``self`` -- the current object.
OUTPUT:
The incidence matrix of the SDP design corresponding to ``self``.
EXAMPLES:
::
sage: from boolean_cayley_graphs.bent_function import BentFunction
sage: bentf = BentFunction([0,0,0,1,0,0,0,1,0,0,0,1,1,1,1,0])
sage: sdp = bentf.sdp_design_matrix()
sage: print(sdp)
[0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0]
[0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 1]
[0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1]
[1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1]
[0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1]
[0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0]
[0 0 1 0 1 1 0 1 0 0 1 0 0 0 1 0]
[1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0]
[0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 1]
[0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0]
[0 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0]
[1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0]
[1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1]
[1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0]
[1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0]
[0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0]
sage: from sage.combinat.designs.incidence_structures import IncidenceStructure
sage: sdp_design = IncidenceStructure(sdp)
sage: sdp_design.is_t_design(return_parameters=True)
(True, (2, 16, 6, 2))
REFERENCES:
.. Dillon and Schatz [DS1987]_, Kantor [Kan1975]_.
"""
dim = self.nvariables()
v = 2**dim
result = matrix(v, v)
dual_self = self.walsh_hadamard_dual()
dual_f = dual_self.extended_translate()
for c in xsrange(v):
result[c, :] = matrix([
self.extended_translate(0, c, dual_f(c))(x) for x in xsrange(v)
])
return result
def small_prime_value(self, Bmax=1000):
r"""
Returns a prime represented by this (primitive positive definite) binary form.
INPUT:
- ``Bmax`` -- a positive bound on the representing integers.
OUTPUT:
A prime number represented by the form.
.. NOTE::
This is a very elementary implementation which just substitutes
values until a prime is found.
EXAMPLES::
sage: [Q.small_prime_value() for Q in BinaryQF_reduced_representatives(-23, primitive_only=True)]
[23, 2, 2]
sage: [Q.small_prime_value() for Q in BinaryQF_reduced_representatives(-47, primitive_only=True)]
[47, 2, 2, 3, 3]
"""
from sage.sets.all import Set
from sage.arith.srange import xsrange
B = 10
while True:
llist = list(Set([self(x,y) for x in xsrange(-B,B) for y in xsrange(B)]))
llist = sorted([l for l in llist if l.is_prime()])
if llist:
return llist[0]
if B >= Bmax:
raise ValueError("Unable to find a prime value of %s" % self)
B += 10
def small_prime_value(self, Bmax=1000):
r"""
Returns a prime represented by this (primitive positive definite) binary form.
INPUT:
- ``Bmax`` -- a positive bound on the representing integers.
OUTPUT:
A prime number represented by the form.
.. NOTE::
This is a very elementary implementation which just substitutes
values until a prime is found.
EXAMPLES::
sage: [Q.small_prime_value() for Q in BinaryQF_reduced_representatives(-23, primitive_only=True)]
[23, 2, 2]
sage: [Q.small_prime_value() for Q in BinaryQF_reduced_representatives(-47, primitive_only=True)]
[47, 2, 2, 3, 3]
"""
from sage.sets.all import Set
from sage.arith.srange import xsrange
B = 10
while True:
llist = list(Set([self(x,y) for x in xsrange(-B,B) for y in xsrange(B)]))
llist = sorted([l for l in llist if l.is_prime()])
if llist:
return llist[0]
if B >= Bmax:
raise ValueError("Unable to find a prime value of %s" % self)
B += 10
def boolean_linear_code(dim, f):
r"""
Return the Boolean linear code corresponding to a Boolean function.
INPUT:
- ``dim`` -- positive integer. The assumed dimension of function ``f``.
- ``f`` -- a Python function that takes a positive integer and returns 0 or 1.
This is assumed to represent a Boolean function on :math:`\mathbb{F}_2^{dim}`
via lexicographical ordering.
OUTPUT:
An object of class ``LinearCode``, representing the Boolean linear code
corresponding to the Boolean function represented by ``f``.
EXAMPLES:
::
sage: from sage.crypto.boolean_function import BooleanFunction
sage: bf = BooleanFunction([0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,1])
sage: dim = bf.nvariables()
sage: from boolean_cayley_graphs.boolean_linear_code import boolean_linear_code
sage: bc = boolean_linear_code(dim, bf)
sage: bc.characteristic_polynomial()
-2/3*x + 2
sage: bc.generator_matrix().echelon_form()
[1 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 1]
REFERENCES:
.. Carlet [Car2010]_.
.. Calderbank and Kantor [CalK1986]_.
.. Ding [Din2015]_ Corollary 10.
"""
v = 2**dim
support = [y for y in xsrange(v) if f(y) == 1]
M = matrix(GF(2),
[[inner(2**k, y) for y in support] for k in xsrange(dim)])
return LinearCode(M)
def least_quadratic_nonresidue(p):
"""
Returns the smallest positive integer quadratic non-residue in Z/pZ for primes p>2.
EXAMPLES::
sage: least_quadratic_nonresidue(5)
2
sage: [least_quadratic_nonresidue(p) for p in prime_range(3,100)]
[2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5]
TESTS:
Raises an error if input is a positive composite integer.
::
sage: least_quadratic_nonresidue(20)
Traceback (most recent call last):
...
ValueError: Oops! p must be a prime number > 2.
Raises an error if input is 2. This is because every integer is a
quadratic residue modulo 2.
::
sage: least_quadratic_nonresidue(2)
Traceback (most recent call last):
...
ValueError: Oops! There are no quadratic non-residues in Z/2Z.
"""
p1 = abs(p)
## Deal with the prime p = 2 and |p| <= 1.
if p1 == 2:
raise ValueError("Oops! There are no quadratic non-residues in Z/2Z.")
if p1 < 2:
raise ValueError("Oops! p must be a prime number > 2.")
## Find the smallest non-residue mod p
## For 7/8 of primes the answer is 2, 3 or 5:
if p % 8 in (3, 5):
return ZZ(2)
if p % 12 in (5, 7):
return ZZ(3)
if p % 5 in (2, 3):
return ZZ(5)
## default case (first needed for p=71):
if not p.is_prime():
raise ValueError("Oops! p must be a prime number > 2.")
from sage.arith.srange import xsrange
for r in xsrange(7, p):
if legendre_symbol(r, p) == -1:
return ZZ(r)
def least_quadratic_nonresidue(p):
"""
Returns the smallest positive integer quadratic non-residue in Z/pZ for primes p>2.
EXAMPLES::
sage: least_quadratic_nonresidue(5)
2
sage: [least_quadratic_nonresidue(p) for p in prime_range(3,100)]
[2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5]
TESTS:
Raises an error if input is a positive composite integer.
::
sage: least_quadratic_nonresidue(20)
Traceback (most recent call last):
...
ValueError: Oops! p must be a prime number > 2.
Raises an error if input is 2. This is because every integer is a
quadratic residue modulo 2.
::
sage: least_quadratic_nonresidue(2)
Traceback (most recent call last):
...
ValueError: Oops! There are no quadratic non-residues in Z/2Z.
"""
p1 = abs(p)
## Deal with the prime p = 2 and |p| <= 1.
if p1 == 2:
raise ValueError("Oops! There are no quadratic non-residues in Z/2Z.")
if p1 < 2:
raise ValueError("Oops! p must be a prime number > 2.")
## Find the smallest non-residue mod p
## For 7/8 of primes the answer is 2, 3 or 5:
if p%8 in (3,5):
return ZZ(2)
if p%12 in (5,7):
return ZZ(3)
if p%5 in (2,3):
return ZZ(5)
## default case (first needed for p=71):
if not p.is_prime():
raise ValueError("Oops! p must be a prime number > 2.")
from sage.arith.srange import xsrange
for r in xsrange(7,p):
if legendre_symbol(r, p) == -1:
return ZZ(r)
def _parametric_plot3d_curve(f, urange, plot_points, **kwds):
r"""
Return a parametric three-dimensional space curve.
This function is used internally by the
:func:`parametric_plot3d` command.
There are two ways this function is invoked by
:func:`parametric_plot3d`.
- ``parametric_plot3d([f_x, f_y, f_z], (u_min,
u_max))``:
`f_x, f_y, f_z` are three functions and
`u_{\min}` and `u_{\max}` are real numbers
- ``parametric_plot3d([f_x, f_y, f_z], (u, u_min,
u_max))``:
`f_x, f_y, f_z` can be viewed as functions of
`u`
INPUT:
- ``f`` - a 3-tuple of functions or expressions, or vector of size 3
- ``urange`` - a 2-tuple (u_min, u_max) or a 3-tuple
(u, u_min, u_max)
- ``plot_points`` - (default: "automatic", which is 75) initial
number of sample points in each parameter; an integer.
EXAMPLES:
We demonstrate each of the two ways of calling this. See
:func:`parametric_plot3d` for many more examples.
We do the first one with a lambda function, which creates a
callable Python function that sends `u` to `u/10`::
sage: parametric_plot3d((sin, cos, lambda u: u/10), (0,20)) # indirect doctest
Graphics3d Object
Now we do the same thing with symbolic expressions::
sage: u = var('u')
sage: parametric_plot3d((sin(u), cos(u), u/10), (u,0,20))
Graphics3d Object
"""
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid(f, [urange], plot_points)
f_x, f_y, f_z = g
w = [(f_x(u), f_y(u), f_z(u))
for u in xsrange(*ranges[0], include_endpoint=True)]
return line3d(w, **kwds)
def _parametric_plot3d_curve(f, urange, plot_points, **kwds):
r"""
Return a parametric three-dimensional space curve.
This function is used internally by the
:func:`parametric_plot3d` command.
There are two ways this function is invoked by
:func:`parametric_plot3d`.
- ``parametric_plot3d([f_x, f_y, f_z], (u_min,
u_max))``:
`f_x, f_y, f_z` are three functions and
`u_{\min}` and `u_{\max}` are real numbers
- ``parametric_plot3d([f_x, f_y, f_z], (u, u_min,
u_max))``:
`f_x, f_y, f_z` can be viewed as functions of
`u`
INPUT:
- ``f`` - a 3-tuple of functions or expressions, or vector of size 3
- ``urange`` - a 2-tuple (u_min, u_max) or a 3-tuple
(u, u_min, u_max)
- ``plot_points`` - (default: "automatic", which is 75) initial
number of sample points in each parameter; an integer.
EXAMPLES:
We demonstrate each of the two ways of calling this. See
:func:`parametric_plot3d` for many more examples.
We do the first one with a lambda function, which creates a
callable Python function that sends `u` to `u/10`::
sage: parametric_plot3d((sin, cos, lambda u: u/10), (0,20)) # indirect doctest
Graphics3d Object
Now we do the same thing with symbolic expressions::
sage: u = var('u')
sage: parametric_plot3d((sin(u), cos(u), u/10), (u,0,20))
Graphics3d Object
"""
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid(f, [urange], plot_points)
f_x, f_y, f_z = g
w = [(f_x(u), f_y(u), f_z(u)) for u in xsrange(*ranges[0], include_endpoint=True)]
return line3d(w, **kwds)
def is_linear_equivalent(self, other, certificate=False):
r"""
Check if there is a linear equivalence between ``self`` and ``other``:
:math:`\mathtt{other}(M x) = \mathtt{self}(x)`,
where M is a GF(2) matrix.
INPUT:
- ``self`` -- the current object.
- ``other`` -- another object of class BooleanFunctionImproved.
- ``certificate`` -- bool (default False). If true, return a GF(2) matrix
that defines the isomorphism.
OUTPUT:
If ``certificate`` is false, a bool value.
If ``certificate`` is true, a tuple consisting of either (False, None)
or (True, M), where M is a GF(2) matrix that defines the equivalence.
EXAMPLES:
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: bf1 = BooleanFunctionImproved([0,1,0,0])
sage: bf2 = BooleanFunctionImproved([0,0,1,0])
sage: bf1.is_linear_equivalent(bf2)
True
sage: bf2.is_linear_equivalent(bf1, certificate=True)
(
[0 1]
True, [1 0]
)
"""
dim = self.nvariables()
self_bg = BooleanGraph(self.cayley_graph())
other_bg = BooleanGraph(other.cayley_graph())
is_linear_isomorphic, M = self_bg.is_linear_isomorphic(
other_bg, certificate=True)
if not is_linear_isomorphic:
return (False, None) if certificate else False
self_et = self.extended_translate()
v = 2**dim
for ix in xsrange(v):
x = vector(GF(2), base2(dim, ix))
if Integer(other(list(M * x))) != self_et(ix):
return (False, None) if certificate else False
return (True, M) if certificate else True
def f(self):
r"""
Returns the set between ``self.start`` and ``self.stop``.
EXAMPLES::
sage: from sage.sets.set_from_iterator import DummyExampleForPicklingTest
sage: d = DummyExampleForPicklingTest()
sage: d.f()
{10, 11, 12, 13, 14, ...}
sage: d.start = 4
sage: d.stop = 200
sage: d.f()
{4, 5, 6, 7, 8, ...}
"""
from sage.arith.srange import xsrange
return xsrange(self.start, self.stop)
def f(self):
r"""
Returns the set between ``self.start`` and ``self.stop``.
EXAMPLES::
sage: from sage.sets.set_from_iterator import DummyExampleForPicklingTest
sage: d = DummyExampleForPicklingTest()
sage: d.f()
{10, 11, 12, 13, 14, ...}
sage: d.start = 4
sage: d.stop = 200
sage: d.f()
{4, 5, 6, 7, 8, ...}
"""
from sage.arith.srange import xsrange
return xsrange(self.start, self.stop)
def solve_integer(self, n):
r"""
Solve `Q(x,y) = n` in integers `x` and `y` where `Q` is this
quadratic form.
INPUT:
- ``Q`` (BinaryQF) -- a positive definite primitive integral
binary quadratic form
- ``n`` (int) -- a positive integer
OUTPUT:
A tuple (x,y) of integers satisfying `Q(x,y) = n` or ``None``
if no such `x` and `y` exist.
EXAMPLES::
sage: Qs = BinaryQF_reduced_representatives(-23,primitive_only=True)
sage: Qs
[x^2 + x*y + 6*y^2, 2*x^2 - x*y + 3*y^2, 2*x^2 + x*y + 3*y^2]
sage: [Q.solve_integer(3) for Q in Qs]
[None, (0, 1), (0, 1)]
sage: [Q.solve_integer(5) for Q in Qs]
[None, None, None]
sage: [Q.solve_integer(6) for Q in Qs]
[(0, 1), (-1, 1), (1, 1)]
"""
a, b, c = self
d = self.discriminant()
if d >= 0 or a <= 0:
raise ValueError("%s is not positive definite" % self)
ad = -d
an4 = 4*a*n
a2 = 2*a
from sage.arith.srange import xsrange
for y in xsrange(0, 1+an4//ad):
z2 = an4 + d*y**2
for z in z2.sqrt(extend=False, all=True):
if a2.divides(z-b*y):
x = (z-b*y)//a2
return (x,y)
return None
def solve_integer(self, n):
r"""
Solve `Q(x,y) = n` in integers `x` and `y` where `Q` is this
quadratic form.
INPUT:
- ``Q`` (BinaryQF) -- a positive definite primitive integral
binary quadratic form
- ``n`` (int) -- a positive integer
OUTPUT:
A tuple (x,y) of integers satisfying `Q(x,y) = n` or ``None``
if no such `x` and `y` exist.
EXAMPLES::
sage: Qs = BinaryQF_reduced_representatives(-23,primitive_only=True)
sage: Qs
[x^2 + x*y + 6*y^2, 2*x^2 - x*y + 3*y^2, 2*x^2 + x*y + 3*y^2]
sage: [Q.solve_integer(3) for Q in Qs]
[None, (0, 1), (0, 1)]
sage: [Q.solve_integer(5) for Q in Qs]
[None, None, None]
sage: [Q.solve_integer(6) for Q in Qs]
[(0, 1), (-1, 1), (1, 1)]
"""
a, b, c = self
d = self.discriminant()
if d >= 0 or a <= 0:
raise ValueError("%s is not positive definite" % self)
ad = -d
an4 = 4*a*n
a2 = 2*a
from sage.arith.srange import xsrange
for y in xsrange(0, 1+an4//ad):
z2 = an4 + d*y**2
for z in z2.sqrt(extend=False, all=True):
if a2.divides(z-b*y):
x = (z-b*y)//a2
return (x,y)
return None
def royle_x_graph():
r"""
Return a strongly regular graph, as described by Royle [Roy2008]_.
INPUT:
None.
OUTPUT:
An object of class ``Graph``, representing Royle's X graph [Roy2008]_.
EXAMPLES:
::
sage: from boolean_cayley_graphs.royle_x_graph import royle_x_graph
sage: g = royle_x_graph()
sage: g.is_strongly_regular()
True
sage: g.is_strongly_regular(parameters=True)
(64, 35, 18, 20)
REFERENCES:
Royle [Roy2008]_.
"""
n = 8
order = 64
vecs = [vector([1] * n)]
for a in Combinations(xsrange(1, n), 4):
vecs.append(vector([-1 if x in a else 1 for x in xsrange(n)]))
for b in Combinations(xsrange(n), 2):
vecs.append(vector([-1 if x in b else 1 for x in xsrange(n)]))
return Graph([(i, j) for i in xsrange(order)
for j in xsrange(i + 1, order) if vecs[i] * vecs[j] == 0])
def is_linear_isomorphic(self, other, certificate=False):
r"""
Check that the two BooleanGraphs ``self`` and ``other`` are isomorphic
and that the isomorphism is given by a GF(2) linear mapping on the
vector space of vertices.
INPUT:
- ``self`` -- the current object.
- ``other`` -- another object of class BooleanFunctionImproved.
- ``certificate`` -- bool (default False). If true, return a GF(2) matrix
that defines the isomorphism.
OUTPUT:
If ``certificate`` is false, a bool value.
If ``certificate`` is true, a tuple consisting of either (False, None)
or (True, M), where M is a GF(2) matrix that defines the isomorphism.
EXAMPLES:
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_graph import BooleanGraph
sage: bf1 = BooleanFunctionImproved([0,1,0,0])
sage: cg1 = BooleanGraph(bf1.cayley_graph())
sage: bf2 = BooleanFunctionImproved([0,0,1,0])
sage: cg2 = BooleanGraph(bf2.cayley_graph())
sage: cg1.is_linear_isomorphic(cg2)
True
sage: cg2.is_linear_isomorphic(cg1, certificate=True)
(
[0 1]
True, [1 0]
)
"""
# Check the isomorphism via canonical labels.
# This is to work around the slow speed of is_isomorphic in some cases.
if self.canonical_label() != other.canonical_label():
return (False, None)
# Obtain the mapping that defines the isomorphism.
is_isomorphic, mapping = self.is_isomorphic(other, certificate=True)
# If self is not isomorphic to other, it is not linear isomorphic.
if not is_isomorphic:
return (False, None) if certificate else False
# Check that the mapping is linear on each pair of basis vectors.
dim = Integer(log(self.order(), 2))
for a in xsrange(dim):
for b in xsrange(a + 1, dim):
if mapping[2**a] ^ mapping[2**b] != mapping[(2**a) ^ (2**b)]:
return (False, None) if certificate else False
# The mapping is linear.
# If the caller does not want a certificate, just return True.
if not certificate:
return True
# Create the G(2) matrix corresponding to the mapping.
mapping_matrix = matrix(
GF(2), [base2(dim, Integer(mapping[2**a]))
for a in xsrange(dim)]).transpose()
return (True, mapping_matrix)
def plot_vector_field(f_g, xrange, yrange, **options):
r"""
``plot_vector_field`` takes two functions of two variables xvar and yvar
(for instance, if the variables are `x` and `y`, take `(f(x,y), g(x,y))`)
and plots vector arrows of the function over the specified ranges, with
xrange being of xvar between xmin and xmax, and yrange similarly
(see below).
``plot_vector_field((f,g), (xvar,xmin,xmax), (yvar,ymin,ymax))``
EXAMPLES:
Plot some vector fields involving sin and cos::
sage: x,y = var('x y')
sage: plot_vector_field((sin(x),cos(y)), (x,-3,3), (y,-3,3))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = plot_vector_field((sin(x),cos(y)), (x,-3,3), (y,-3,3))
sphinx_plot(g)
::
sage: plot_vector_field((y,(cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = plot_vector_field((y,(cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi))
sphinx_plot(g)
Plot a gradient field::
sage: u, v = var('u v')
sage: f = exp(-(u^2 + v^2))
sage: plot_vector_field(f.gradient(), (u,-2,2), (v,-2,2), color='blue')
Graphics object consisting of 1 graphics primitive
.. PLOT::
u, v = var('u v')
f = exp(-(u**2 + v**2))
g = plot_vector_field(f.gradient(), (u,-2,2), (v,-2,2), color='blue')
sphinx_plot(g)
Plot two orthogonal vector fields::
sage: x,y = var('x,y')
sage: a = plot_vector_field((x,y), (x,-3,3), (y,-3,3), color='blue')
sage: b = plot_vector_field((y,-x), (x,-3,3), (y,-3,3), color='red')
sage: show(a + b)
.. PLOT::
x,y = var('x,y')
a = plot_vector_field((x,y), (x,-3,3), (y,-3,3), color='blue')
b = plot_vector_field((y,-x), (x,-3,3), (y,-3,3), color='red')
sphinx_plot(a + b)
We ignore function values that are infinite or NaN::
sage: x,y = var('x,y')
sage: plot_vector_field((-x/sqrt(x^2+y^2),-y/sqrt(x^2+y^2)), (x,-10,10), (y,-10,10))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
g = plot_vector_field((-x/sqrt(x**2+y**2),-y/sqrt(x**2+y**2)), (x,-10,10), (y,-10,10))
sphinx_plot(g)
::
sage: x,y = var('x,y')
sage: plot_vector_field((-x/sqrt(x+y),-y/sqrt(x+y)), (x,-10, 10), (y,-10,10))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
g = plot_vector_field((-x/sqrt(x+y),-y/sqrt(x+y)), (x,-10,10), (y,-10,10))
sphinx_plot(g)
Extra options will get passed on to show(), as long as they are valid::
sage: plot_vector_field((x,y), (x,-2,2), (y,-2,2), xmax=10)
Graphics object consisting of 1 graphics primitive
sage: plot_vector_field((x,y), (x,-2,2), (y,-2,2)).show(xmax=10) # These are equivalent
.. PLOT::
x,y = var('x,y')
g = plot_vector_field((x,y), (x,-2,2), (y,-2,2), xmax=10)
sphinx_plot(g)
"""
(f,g) = f_g
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
z, ranges = setup_for_eval_on_grid([f,g], [xrange,yrange], options['plot_points'])
f, g = z
xpos_array, ypos_array, xvec_array, yvec_array = [], [], [], []
for x in xsrange(*ranges[0], include_endpoint=True):
for y in xsrange(*ranges[1], include_endpoint=True):
xpos_array.append(x)
ypos_array.append(y)
xvec_array.append(f(x, y))
yvec_array.append(g(x, y))
import numpy
xvec_array = numpy.ma.masked_invalid(numpy.array(xvec_array, dtype=float))
yvec_array = numpy.ma.masked_invalid(numpy.array(yvec_array, dtype=float))
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options))
g.add_primitive(PlotField(xpos_array, ypos_array,
xvec_array, yvec_array, options))
return g
def _render_on_subplot(self, subplot):
"""
TESTS:
A somewhat random plot, but fun to look at::
sage: x,y = var('x,y')
sage: contour_plot(x^2-y^3+10*sin(x*y), (x, -4, 4), (y, -4, 4),plot_points=121,cmap='hsv')
Graphics object consisting of 1 graphics primitive
"""
from sage.rings.integer import Integer
options = self.options()
fill = options['fill']
contours = options['contours']
if 'cmap' in options:
cmap = get_cmap(options['cmap'])
elif fill or contours is None:
cmap = get_cmap('gray')
else:
if isinstance(contours, (int, Integer)):
cmap = get_cmap([(i, i, i)
for i in xsrange(0, 1, 1 / contours)])
else:
l = Integer(len(contours))
cmap = get_cmap([(i, i, i) for i in xsrange(0, 1, 1 / l)])
x0, x1 = float(self.xrange[0]), float(self.xrange[1])
y0, y1 = float(self.yrange[0]), float(self.yrange[1])
if isinstance(contours, (int, Integer)):
contours = int(contours)
CSF = None
if fill:
if contours is None:
CSF = subplot.contourf(self.xy_data_array,
cmap=cmap,
extent=(x0, x1, y0, y1),
label=options['legend_label'])
else:
CSF = subplot.contourf(self.xy_data_array,
contours,
cmap=cmap,
extent=(x0, x1, y0, y1),
extend='both',
label=options['legend_label'])
linewidths = options.get('linewidths', None)
if isinstance(linewidths, (int, Integer)):
linewidths = int(linewidths)
elif isinstance(linewidths, (list, tuple)):
linewidths = tuple(int(x) for x in linewidths)
from sage.plot.misc import get_matplotlib_linestyle
linestyles = options.get('linestyles', None)
if isinstance(linestyles, (list, tuple)):
linestyles = [
get_matplotlib_linestyle(l, 'long') for l in linestyles
]
else:
linestyles = get_matplotlib_linestyle(linestyles, 'long')
if contours is None:
CS = subplot.contour(self.xy_data_array,
cmap=cmap,
extent=(x0, x1, y0, y1),
linewidths=linewidths,
linestyles=linestyles,
label=options['legend_label'])
else:
CS = subplot.contour(self.xy_data_array,
contours,
cmap=cmap,
extent=(x0, x1, y0, y1),
linewidths=linewidths,
linestyles=linestyles,
label=options['legend_label'])
if options.get('labels', False):
label_options = options['label_options']
label_options['fontsize'] = int(label_options['fontsize'])
if fill and label_options is None:
label_options['inline'] = False
subplot.clabel(CS, **label_options)
if options.get('colorbar', False):
colorbar_options = options['colorbar_options']
from matplotlib import colorbar
cax, kwds = colorbar.make_axes_gridspec(subplot,
**colorbar_options)
if CSF is None:
cb = colorbar.Colorbar(cax, CS, **kwds)
else:
cb = colorbar.Colorbar(cax, CSF, **kwds)
cb.add_lines(CS)
def BinaryQF_reduced_representatives(D, primitive_only=False):
r"""
Return representatives for the classes of binary quadratic forms
of discriminant `D`.
INPUT:
- ``D`` -- (integer) a discriminant
- ``primitive_only`` -- (boolean, default True): if True, only
return primitive forms.
OUTPUT:
(list) A lexicographically-ordered list of inequivalent reduced
representatives for the equivalence classes of binary quadratic
forms of discriminant `D`. If ``primitive_only`` is ``True`` then
imprimitive forms (which only exist when `D` is not fundamental) are
omitted; otherwise they are included.
EXAMPLES::
sage: BinaryQF_reduced_representatives(-4)
[x^2 + y^2]
sage: BinaryQF_reduced_representatives(-163)
[x^2 + x*y + 41*y^2]
sage: BinaryQF_reduced_representatives(-12)
[x^2 + 3*y^2, 2*x^2 + 2*x*y + 2*y^2]
sage: BinaryQF_reduced_representatives(-16)
[x^2 + 4*y^2, 2*x^2 + 2*y^2]
sage: BinaryQF_reduced_representatives(-63)
[x^2 + x*y + 16*y^2, 2*x^2 - x*y + 8*y^2, 2*x^2 + x*y + 8*y^2, 3*x^2 + 3*x*y + 6*y^2, 4*x^2 + x*y + 4*y^2]
The number of inequivalent reduced binary forms with a fixed negative
fundamental discriminant D is the class number of the quadratic field
`\QQ(\sqrt{D})`::
sage: len(BinaryQF_reduced_representatives(-13*4))
2
sage: QuadraticField(-13*4, 'a').class_number()
2
sage: p=next_prime(2^20); p
1048583
sage: len(BinaryQF_reduced_representatives(-p))
689
sage: QuadraticField(-p, 'a').class_number()
689
sage: BinaryQF_reduced_representatives(-23*9)
[x^2 + x*y + 52*y^2,
2*x^2 - x*y + 26*y^2,
2*x^2 + x*y + 26*y^2,
3*x^2 + 3*x*y + 18*y^2,
4*x^2 - x*y + 13*y^2,
4*x^2 + x*y + 13*y^2,
6*x^2 - 3*x*y + 9*y^2,
6*x^2 + 3*x*y + 9*y^2,
8*x^2 + 7*x*y + 8*y^2]
sage: BinaryQF_reduced_representatives(-23*9, primitive_only=True)
[x^2 + x*y + 52*y^2,
2*x^2 - x*y + 26*y^2,
2*x^2 + x*y + 26*y^2,
4*x^2 - x*y + 13*y^2,
4*x^2 + x*y + 13*y^2,
8*x^2 + 7*x*y + 8*y^2]
TESTS::
sage: BinaryQF_reduced_representatives(73)
[-6*x^2 + 5*x*y + 2*y^2,
-6*x^2 + 7*x*y + y^2,
-4*x^2 + 3*x*y + 4*y^2,
-4*x^2 + 3*x*y + 4*y^2,
-4*x^2 + 5*x*y + 3*y^2,
-3*x^2 + 5*x*y + 4*y^2,
-3*x^2 + 7*x*y + 2*y^2,
-2*x^2 + 5*x*y + 6*y^2,
-2*x^2 + 7*x*y + 3*y^2,
-x^2 + 7*x*y + 6*y^2,
x^2 + 7*x*y - 6*y^2,
2*x^2 + 5*x*y - 6*y^2,
2*x^2 + 7*x*y - 3*y^2,
3*x^2 + 5*x*y - 4*y^2,
3*x^2 + 7*x*y - 2*y^2,
4*x^2 + 3*x*y - 4*y^2,
4*x^2 + 3*x*y - 4*y^2,
4*x^2 + 5*x*y - 3*y^2,
6*x^2 + 5*x*y - 2*y^2,
6*x^2 + 7*x*y - y^2]
sage: BinaryQF_reduced_representatives(76, primitive_only=True)
[-5*x^2 + 4*x*y + 3*y^2,
-5*x^2 + 6*x*y + 2*y^2,
-3*x^2 + 4*x*y + 5*y^2,
-3*x^2 + 8*x*y + y^2,
-2*x^2 + 6*x*y + 5*y^2,
-x^2 + 8*x*y + 3*y^2,
x^2 + 8*x*y - 3*y^2,
2*x^2 + 6*x*y - 5*y^2,
3*x^2 + 4*x*y - 5*y^2,
3*x^2 + 8*x*y - y^2,
5*x^2 + 4*x*y - 3*y^2,
5*x^2 + 6*x*y - 2*y^2]
Check that the primitive_only keyword does something::
sage: BinaryQF_reduced_representatives(4*5, primitive_only=True)
[-x^2 + 4*x*y + y^2,
-x^2 + 4*x*y + y^2,
x^2 + 4*x*y - y^2,
x^2 + 4*x*y - y^2]
sage: BinaryQF_reduced_representatives(4*5, primitive_only=False)
[-2*x^2 + 2*x*y + 2*y^2,
-2*x^2 + 2*x*y + 2*y^2,
-x^2 + 4*x*y + y^2,
-x^2 + 4*x*y + y^2,
x^2 + 4*x*y - y^2,
x^2 + 4*x*y - y^2,
2*x^2 + 2*x*y - 2*y^2,
2*x^2 + 2*x*y - 2*y^2]
"""
D = ZZ(D)
# For a fundamental discriminant all forms are primitive so we need not check:
if primitive_only:
primitive_only = not is_fundamental_discriminant(D)
form_list = []
from sage.arith.srange import xsrange
D4 = D % 4
if D4 == 2 or D4 == 3:
raise ValueError("%s is not a discriminant" % D)
if D > 0: # Indefinite
# We follow the description of Buchmann/Vollmer 6.7.1
if D.is_square():
# Buchmann/Vollmer 6.7.1. require D squarefree.
raise ValueError("%s is a square" % D)
sqrt_d = D.sqrt(prec=53)
for b in xsrange(1, sqrt_d.floor() + 1):
if (D - b) % 2 != 0:
continue
A = (D - b**2) / 4
Low_a = ((sqrt_d - b) / 2).ceil()
High_a = (A.sqrt(prec=53)).floor()
for a in xsrange(Low_a, High_a + 1):
if a == 0:
continue
c = -A / a
if c in ZZ:
if (not primitive_only) or gcd([a, b, c]) == 1:
Q = BinaryQF(a, b, c)
Q1 = BinaryQF(-a, b, -c)
Q2 = BinaryQF(c, b, a)
Q3 = BinaryQF(-c, b, -a)
form_list.append(Q)
form_list.append(Q1)
form_list.append(Q2)
form_list.append(Q3)
else: # Definite
# Only iterate over positive a and over b of the same
# parity as D such that 4a^2 + D <= b^2 <= a^2
for a in xsrange(1, 1 + ((-D) // 3).isqrt()):
a4 = 4 * a
s = D + a * a4
w = 1 + (s - 1).isqrt() if s > 0 else 0
if w % 2 != D % 2: w += 1
for b in xsrange(w, a + 1, 2):
t = b * b - D
if t % a4 == 0:
c = t // a4
if (not primitive_only) or gcd([a, b, c]) == 1:
if b > 0 and a > b and c > a:
form_list.append(BinaryQF([a, -b, c]))
form_list.append(BinaryQF([a, b, c]))
form_list.sort()
return form_list
def plot_vector_field(f_g, xrange, yrange, **options):
r"""
``plot_vector_field`` takes two functions of two variables xvar and yvar
(for instance, if the variables are `x` and `y`, take `(f(x,y), g(x,y))`)
and plots vector arrows of the function over the specified ranges, with
xrange being of xvar between xmin and xmax, and yrange similarly
(see below).
``plot_vector_field((f,g), (xvar,xmin,xmax), (yvar,ymin,ymax))``
EXAMPLES:
Plot some vector fields involving sin and cos::
sage: x,y = var('x y')
sage: plot_vector_field((sin(x),cos(y)), (x,-3,3), (y,-3,3))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = plot_vector_field((sin(x),cos(y)), (x,-3,3), (y,-3,3))
sphinx_plot(g)
::
sage: plot_vector_field((y,(cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = plot_vector_field((y,(cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi))
sphinx_plot(g)
Plot a gradient field::
sage: u, v = var('u v')
sage: f = exp(-(u^2 + v^2))
sage: plot_vector_field(f.gradient(), (u,-2,2), (v,-2,2), color='blue')
Graphics object consisting of 1 graphics primitive
.. PLOT::
u, v = var('u v')
f = exp(-(u**2 + v**2))
g = plot_vector_field(f.gradient(), (u,-2,2), (v,-2,2), color='blue')
sphinx_plot(g)
Plot two orthogonal vector fields::
sage: x,y = var('x,y')
sage: a = plot_vector_field((x,y), (x,-3,3), (y,-3,3), color='blue')
sage: b = plot_vector_field((y,-x), (x,-3,3), (y,-3,3), color='red')
sage: show(a + b)
.. PLOT::
x,y = var('x,y')
a = plot_vector_field((x,y), (x,-3,3), (y,-3,3), color='blue')
b = plot_vector_field((y,-x), (x,-3,3), (y,-3,3), color='red')
sphinx_plot(a + b)
We ignore function values that are infinite or NaN::
sage: x,y = var('x,y')
sage: plot_vector_field((-x/sqrt(x^2+y^2),-y/sqrt(x^2+y^2)), (x,-10,10), (y,-10,10))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
g = plot_vector_field((-x/sqrt(x**2+y**2),-y/sqrt(x**2+y**2)), (x,-10,10), (y,-10,10))
sphinx_plot(g)
::
sage: x,y = var('x,y')
sage: plot_vector_field((-x/sqrt(x+y),-y/sqrt(x+y)), (x,-10, 10), (y,-10,10))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
g = plot_vector_field((-x/sqrt(x+y),-y/sqrt(x+y)), (x,-10,10), (y,-10,10))
sphinx_plot(g)
Extra options will get passed on to show(), as long as they are valid::
sage: plot_vector_field((x,y), (x,-2,2), (y,-2,2), xmax=10)
Graphics object consisting of 1 graphics primitive
sage: plot_vector_field((x,y), (x,-2,2), (y,-2,2)).show(xmax=10) # These are equivalent
.. PLOT::
x,y = var('x,y')
g = plot_vector_field((x,y), (x,-2,2), (y,-2,2), xmax=10)
sphinx_plot(g)
"""
(f, g) = f_g
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
z, ranges = setup_for_eval_on_grid([f, g], [xrange, yrange],
options['plot_points'])
f, g = z
xpos_array, ypos_array, xvec_array, yvec_array = [], [], [], []
for x in xsrange(*ranges[0], include_endpoint=True):
for y in xsrange(*ranges[1], include_endpoint=True):
xpos_array.append(x)
ypos_array.append(y)
xvec_array.append(f(x, y))
yvec_array.append(g(x, y))
import numpy
xvec_array = numpy.ma.masked_invalid(numpy.array(xvec_array, dtype=float))
yvec_array = numpy.ma.masked_invalid(numpy.array(yvec_array, dtype=float))
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options))
g.add_primitive(
PlotField(xpos_array, ypos_array, xvec_array, yvec_array, options))
return g
def contour_plot(f, xrange, yrange, **options):
r"""
``contour_plot`` takes a function of two variables, `f(x,y)`
and plots contour lines of the function over the specified
``xrange`` and ``yrange`` as demonstrated below.
``contour_plot(f, (xmin, xmax), (ymin, ymax), ...)``
INPUT:
- ``f`` -- a function of two variables
- ``(xmin, xmax)`` -- 2-tuple, the range of ``x`` values OR 3-tuple
``(x,xmin,xmax)``
- ``(ymin, ymax)`` -- 2-tuple, the range of ``y`` values OR 3-tuple
``(y,ymin,ymax)``
The following inputs must all be passed in as named parameters:
- ``plot_points`` -- integer (default: 100); number of points to plot
in each direction of the grid. For old computers, 25 is fine, but
should not be used to verify specific intersection points.
- ``fill`` -- bool (default: ``True``), whether to color in the area
between contour lines
- ``cmap`` -- a colormap (default: ``'gray'``), the name of
a predefined colormap, a list of colors or an instance of a matplotlib
Colormap. Type: ``import matplotlib.cm; matplotlib.cm.datad.keys()``
for available colormap names.
- ``contours`` -- integer or list of numbers (default: ``None``):
If a list of numbers is given, then this specifies the contour levels
to use. If an integer is given, then this many contour lines are
used, but the exact levels are determined automatically. If ``None``
is passed (or the option is not given), then the number of contour
lines is determined automatically, and is usually about 5.
- ``linewidths`` -- integer or list of integer (default: None), if
a single integer all levels will be of the width given,
otherwise the levels will be plotted with the width in the order
given. If the list is shorter than the number of contours, then
the widths will be repeated cyclically.
- ``linestyles`` -- string or list of strings (default: None), the
style of the lines to be plotted, one of: ``"solid"``, ``"dashed"``,
``"dashdot"``, ``"dotted"``, respectively ``"-"``, ``"--"``,
``"-."``, ``":"``. If the list is shorter than the number of
contours, then the styles will be repeated cyclically.
- ``labels`` -- boolean (default: False) Show level labels or not.
The following options are to adjust the style and placement of
labels, they have no effect if no labels are shown.
- ``label_fontsize`` -- integer (default: 9), the font size of the labels.
- ``label_colors`` -- string or sequence of colors (default:
None) If a string, gives the name of a single color with which
to draw all labels. If a sequence, gives the colors of the
labels. A color is a string giving the name of one or a
3-tuple of floats.
- ``label_inline`` -- boolean (default: False if fill is True,
otherwise True), controls whether the underlying contour is
removed or not.
- ``label_inline_spacing`` -- integer (default: 3), When inline,
this is the amount of contour that is removed from each side,
in pixels.
- ``label_fmt`` -- a format string (default: "%1.2f"), this is
used to get the label text from the level. This can also be a
dictionary with the contour levels as keys and corresponding
text string labels as values. It can also be any callable which
returns a string when called with a numeric contour level.
- ``colorbar`` -- boolean (default: False) Show a colorbar or not.
The following options are to adjust the style and placement of
colorbars. They have no effect if a colorbar is not shown.
- ``colorbar_orientation`` -- string (default: 'vertical'),
controls placement of the colorbar, can be either 'vertical'
or 'horizontal'
- ``colorbar_format`` -- a format string, this is used to format
the colorbar labels.
- ``colorbar_spacing`` -- string (default: 'proportional'). If
'proportional', make the contour divisions proportional to
values. If 'uniform', space the colorbar divisions uniformly,
without regard for numeric values.
- ``legend_label`` -- the label for this item in the legend
- ``region`` - (default: None) If region is given, it must be a function
of two variables. Only segments of the surface where region(x,y) returns a
number >0 will be included in the plot.
EXAMPLES:
Here we plot a simple function of two variables. Note that
since the input function is an expression, we need to explicitly
declare the variables in 3-tuples for the range::
sage: x,y = var('x,y')
sage: contour_plot(cos(x^2+y^2), (x, -4, 4), (y, -4, 4))
Graphics object consisting of 1 graphics primitive
Here we change the ranges and add some options::
sage: x,y = var('x,y')
sage: contour_plot((x^2)*cos(x*y), (x, -10, 5), (y, -5, 5), fill=False, plot_points=150)
Graphics object consisting of 1 graphics primitive
An even more complicated plot::
sage: x,y = var('x,y')
sage: contour_plot(sin(x^2 + y^2)*cos(x)*sin(y), (x, -4, 4), (y, -4, 4),plot_points=150)
Graphics object consisting of 1 graphics primitive
Some elliptic curves, but with symbolic endpoints. In the first
example, the plot is rotated 90 degrees because we switch the
variables `x`, `y`::
sage: x,y = var('x,y')
sage: contour_plot(y^2 + 1 - x^3 - x, (y,-pi,pi), (x,-pi,pi))
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
We can play with the contour levels::
sage: x,y = var('x,y')
sage: f(x,y) = x^2 + y^2
sage: contour_plot(f, (-2, 2), (-2, 2))
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (-2, 2), (-2, 2), contours=2, cmap=[(1,0,0), (0,1,0), (0,0,1)])
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (-2, 2), (-2, 2), contours=(0.1, 1.0, 1.2, 1.4), cmap='hsv')
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (-2, 2), (-2, 2), contours=(1.0,), fill=False)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(x-y^2,(x,-5,5),(y,-3,3),contours=[-4,0,1])
Graphics object consisting of 1 graphics primitive
We can change the style of the lines::
sage: contour_plot(f, (-2,2), (-2,2), fill=False, linewidths=10)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (-2,2), (-2,2), fill=False, linestyles='dashdot')
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(x^2-y^2,(x,-3,3),(y,-3,3),contours=[0,1,2,3,4],\
... linewidths=[1,5],linestyles=['solid','dashed'],fill=False)
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(x^2-y^2,(x,-3,3),(y,-3,3),contours=[0,1,2,3,4],\
... linewidths=[1,5],linestyles=['solid','dashed'])
sage: P
Graphics object consisting of 1 graphics primitive
sage: P=contour_plot(x^2-y^2,(x,-3,3),(y,-3,3),contours=[0,1,2,3,4],\
... linewidths=[1,5],linestyles=['-',':'])
sage: P
Graphics object consisting of 1 graphics primitive
We can add labels and play with them::
sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv', labels=True)
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv',\
... labels=True, label_fmt="%1.0f", label_colors='black')
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv',labels=True,\
... contours=[-4,0,4], label_fmt={-4:"low", 0:"medium", 4: "hi"}, label_colors='black')
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv',labels=True,\
... contours=[-4,0,4], label_fmt=lambda x: "$z=%s$"%x, label_colors='black', label_inline=True, \
... label_fontsize=12)
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), \
... fill=False, cmap='hsv', labels=True, label_fontsize=18)
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), \
... fill=False, cmap='hsv', labels=True, label_inline_spacing=1)
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P= contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), \
... fill=False, cmap='hsv', labels=True, label_inline=False)
sage: P
Graphics object consisting of 1 graphics primitive
We can change the color of the labels if so desired::
sage: contour_plot(f, (-2,2), (-2,2), labels=True, label_colors='red')
Graphics object consisting of 1 graphics primitive
We can add a colorbar as well::
sage: f(x,y)=x^2-y^2
sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True,colorbar_orientation='horizontal')
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4],colorbar=True)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4],colorbar=True,colorbar_spacing='uniform')
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[0,2,3,6],colorbar=True,colorbar_format='%.3f')
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), labels=True,label_colors='red',contours=[0,2,3,6],colorbar=True)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), cmap='winter', contours=20, fill=False, colorbar=True)
Graphics object consisting of 1 graphics primitive
This should plot concentric circles centered at the origin::
sage: x,y = var('x,y')
sage: contour_plot(x^2+y^2-2,(x,-1,1), (y,-1,1))
Graphics object consisting of 1 graphics primitive
Extra options will get passed on to show(), as long as they are valid::
sage: f(x, y) = cos(x) + sin(y)
sage: contour_plot(f, (0, pi), (0, pi), axes=True)
Graphics object consisting of 1 graphics primitive
One can also plot over a reduced region::
sage: contour_plot(x**2-y**2, (x,-2, 2), (y,-2, 2),region=x-y,plot_points=300)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (0, pi), (0, pi)).show(axes=True) # These are equivalent
Note that with ``fill=False`` and grayscale contours, there is the
possibility of confusion between the contours and the axes, so use
``fill=False`` together with ``axes=True`` with caution::
sage: contour_plot(f, (-pi, pi), (-pi, pi), fill=False, axes=True)
Graphics object consisting of 1 graphics primitive
TESTS:
To check that :trac:`5221` is fixed, note that this has three curves, not two::
sage: x,y = var('x,y')
sage: contour_plot(x-y^2,(x,-5,5),(y,-3,3),contours=[-4,-2,0], fill=False)
Graphics object consisting of 1 graphics primitive
"""
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
region = options.pop('region')
ev = [f] if region is None else [f, region]
F, ranges = setup_for_eval_on_grid(ev, [xrange, yrange],
options['plot_points'])
g = F[0]
xrange, yrange = [r[:2] for r in ranges]
xy_data_array = [[
g(x, y) for x in xsrange(*ranges[0], include_endpoint=True)
] for y in xsrange(*ranges[1], include_endpoint=True)]
if region is not None:
import numpy
xy_data_array = numpy.ma.asarray(xy_data_array, dtype=float)
m = F[1]
mask = numpy.asarray([[
m(x, y) <= 0 for x in xsrange(*ranges[0], include_endpoint=True)
] for y in xsrange(*ranges[1], include_endpoint=True)],
dtype=bool)
xy_data_array[mask] = numpy.ma.masked
g = Graphics()
# Reset aspect_ratio to 'automatic' in case scale is 'semilog[xy]'.
# Otherwise matplotlib complains.
scale = options.get('scale', None)
if isinstance(scale, (list, tuple)):
scale = scale[0]
if scale == 'semilogy' or scale == 'semilogx':
options['aspect_ratio'] = 'automatic'
g._set_extra_kwds(
Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax']))
g.add_primitive(ContourPlot(xy_data_array, xrange, yrange, options))
return g
def region_plot(f, xrange, yrange, plot_points, incol, outcol, bordercol, borderstyle, borderwidth, alpha, **options):
r"""
``region_plot`` takes a boolean function of two variables, `f(x,y)`
and plots the region where f is True over the specified
``xrange`` and ``yrange`` as demonstrated below.
``region_plot(f, (xmin, xmax), (ymin, ymax), ...)``
INPUT:
- ``f`` -- a boolean function or a list of boolean functions of two variables
- ``(xmin, xmax)`` -- 2-tuple, the range of ``x`` values OR 3-tuple
``(x,xmin,xmax)``
- ``(ymin, ymax)`` -- 2-tuple, the range of ``y`` values OR 3-tuple
``(y,ymin,ymax)``
- ``plot_points`` -- integer (default: 100); number of points to plot
in each direction of the grid
- ``incol`` -- a color (default: ``'blue'``), the color inside the region
- ``outcol`` -- a color (default: ``None``), the color of the outside
of the region
If any of these options are specified, the border will be shown as indicated,
otherwise it is only implicit (with color ``incol``) as the border of the
inside of the region.
- ``bordercol`` -- a color (default: ``None``), the color of the border
(``'black'`` if ``borderwidth`` or ``borderstyle`` is specified but not ``bordercol``)
- ``borderstyle`` -- string (default: 'solid'), one of ``'solid'``,
``'dashed'``, ``'dotted'``, ``'dashdot'``, respectively ``'-'``,
``'--'``, ``':'``, ``'-.'``.
- ``borderwidth`` -- integer (default: None), the width of the border in pixels
- ``alpha`` -- (default: 1) How transparent the fill is. A number between 0 and 1.
- ``legend_label`` -- the label for this item in the legend
- ``base`` - (default: 10) the base of the logarithm if
a logarithmic scale is set. This must be greater than 1. The base
can be also given as a list or tuple ``(basex, basey)``.
``basex`` sets the base of the logarithm along the horizontal
axis and ``basey`` sets the base along the vertical axis.
- ``scale`` -- (default: ``"linear"``) string. The scale of the axes.
Possible values are ``"linear"``, ``"loglog"``, ``"semilogx"``,
``"semilogy"``.
The scale can be also be given as single argument that is a list
or tuple ``(scale, base)`` or ``(scale, basex, basey)``.
The ``"loglog"`` scale sets both the horizontal and vertical axes to
logarithmic scale. The ``"semilogx"`` scale sets the horizontal axis
to logarithmic scale. The ``"semilogy"`` scale sets the vertical axis
to logarithmic scale. The ``"linear"`` scale is the default value
when :class:`~sage.plot.graphics.Graphics` is initialized.
EXAMPLES:
Here we plot a simple function of two variables::
sage: x,y = var('x,y')
sage: region_plot(cos(x^2+y^2) <= 0, (x, -3, 3), (y, -3, 3))
Graphics object consisting of 1 graphics primitive
Here we play with the colors::
sage: region_plot(x^2+y^3 < 2, (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray')
Graphics object consisting of 2 graphics primitives
An even more complicated plot, with dashed borders::
sage: region_plot(sin(x)*sin(y) >= 1/4, (x,-10,10), (y,-10,10), incol='yellow', bordercol='black', borderstyle='dashed', plot_points=250)
Graphics object consisting of 2 graphics primitives
A disk centered at the origin::
sage: region_plot(x^2+y^2<1, (x,-1,1), (y,-1,1))
Graphics object consisting of 1 graphics primitive
A plot with more than one condition (all conditions must be true for the statement to be true)::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2))
Graphics object consisting of 1 graphics primitive
Since it doesn't look very good, let's increase ``plot_points``::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2), plot_points=400)
Graphics object consisting of 1 graphics primitive
To get plots where only one condition needs to be true, use a function.
Using lambda functions, we definitely need the extra ``plot_points``::
sage: region_plot(lambda x,y: x^2+y^2<1 or x<y, (x,-2,2), (y,-2,2), plot_points=400)
Graphics object consisting of 1 graphics primitive
The first quadrant of the unit circle::
sage: region_plot([y>0, x>0, x^2+y^2<1], (x,-1.1, 1.1), (y,-1.1, 1.1), plot_points = 400)
Graphics object consisting of 1 graphics primitive
Here is another plot, with a huge border::
sage: region_plot(x*(x-1)*(x+1)+y^2<0, (x, -3, 2), (y, -3, 3), incol='lightblue', bordercol='gray', borderwidth=10, plot_points=50)
Graphics object consisting of 2 graphics primitives
If we want to keep only the region where x is positive::
sage: region_plot([x*(x-1)*(x+1)+y^2<0, x>-1], (x, -3, 2), (y, -3, 3), incol='lightblue', plot_points=50)
Graphics object consisting of 1 graphics primitive
Here we have a cut circle::
sage: region_plot([x^2+y^2<4, x>-1], (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray', plot_points=200)
Graphics object consisting of 2 graphics primitives
The first variable range corresponds to the horizontal axis and
the second variable range corresponds to the vertical axis::
sage: s,t=var('s,t')
sage: region_plot(s>0,(t,-2,2),(s,-2,2))
Graphics object consisting of 1 graphics primitive
::
sage: region_plot(s>0,(s,-2,2),(t,-2,2))
Graphics object consisting of 1 graphics primitive
An example of a region plot in 'loglog' scale::
sage: region_plot(x^2+y^2<100, (x,1,10), (y,1,10), scale='loglog')
Graphics object consisting of 1 graphics primitive
TESTS:
To check that :trac:`16907` is fixed::
sage: x, y = var('x, y')
sage: disc1 = region_plot(x^2+y^2 < 1, (x, -1, 1), (y, -1, 1), alpha=0.5)
sage: disc2 = region_plot((x-0.7)^2+(y-0.7)^2 < 0.5, (x, -2, 2), (y, -2, 2), incol='red', alpha=0.5)
sage: disc1 + disc2
Graphics object consisting of 2 graphics primitives
To check that :trac:`18286` is fixed::
sage: x, y = var('x, y')
sage: region_plot([x == 0], (x, -1, 1), (y, -1, 1))
Graphics object consisting of 1 graphics primitive
sage: region_plot([x^2+y^2==1, x<y], (x, -1, 1), (y, -1, 1))
Graphics object consisting of 1 graphics primitive
"""
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
from sage.symbolic.expression import is_Expression
from warnings import warn
import numpy
if not isinstance(f, (list, tuple)):
f = [f]
feqs = [equify(g) for g in f if is_Expression(g) and g.operator() is operator.eq and not equify(g).is_zero()]
f = [equify(g) for g in f if not (is_Expression(g) and g.operator() is operator.eq)]
neqs = len(feqs)
if neqs > 1:
warn("There are at least 2 equations; If the region is degenerated to points, plotting might show nothing.")
feqs = [sum([fn**2 for fn in feqs])]
neqs = 1
if neqs and not bordercol:
bordercol = incol
if not f:
return implicit_plot(feqs[0], xrange, yrange, plot_points=plot_points, fill=False, \
linewidth=borderwidth, linestyle=borderstyle, color=bordercol, **options)
f_all, ranges = setup_for_eval_on_grid(feqs + f, [xrange, yrange], plot_points)
xrange,yrange=[r[:2] for r in ranges]
xy_data_arrays = numpy.asarray([[[func(x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)]
for func in f_all[neqs::]],dtype=float)
xy_data_array=numpy.abs(xy_data_arrays.prod(axis=0))
# Now we need to set entries to negative iff all
# functions were negative at that point.
neg_indices = (xy_data_arrays<0).all(axis=0)
xy_data_array[neg_indices]=-xy_data_array[neg_indices]
from matplotlib.colors import ListedColormap
incol = rgbcolor(incol)
if outcol:
outcol = rgbcolor(outcol)
cmap = ListedColormap([incol, outcol])
cmap.set_over(outcol, alpha=alpha)
else:
outcol = rgbcolor('white')
cmap = ListedColormap([incol, outcol])
cmap.set_over(outcol, alpha=0)
cmap.set_under(incol, alpha=alpha)
g = Graphics()
# Reset aspect_ratio to 'automatic' in case scale is 'semilog[xy]'.
# Otherwise matplotlib complains.
scale = options.get('scale', None)
if isinstance(scale, (list, tuple)):
scale = scale[0]
if scale == 'semilogy' or scale == 'semilogx':
options['aspect_ratio'] = 'automatic'
g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax']))
if neqs == 0:
g.add_primitive(ContourPlot(xy_data_array, xrange,yrange,
dict(contours=[-1e-20, 0, 1e-20], cmap=cmap, fill=True, **options)))
else:
mask = numpy.asarray([[elt > 0 for elt in rows] for rows in xy_data_array], dtype=bool)
xy_data_array = numpy.asarray([[f_all[0](x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)], dtype=float)
xy_data_array[mask] = None
if bordercol or borderstyle or borderwidth:
cmap = [rgbcolor(bordercol)] if bordercol else ['black']
linestyles = [borderstyle] if borderstyle else None
linewidths = [borderwidth] if borderwidth else None
g.add_primitive(ContourPlot(xy_data_array, xrange, yrange,
dict(linestyles=linestyles, linewidths=linewidths,
contours=[0], cmap=[bordercol], fill=False, **options)))
return g
def region_plot(f, xrange, yrange, plot_points, incol, outcol, bordercol,
borderstyle, borderwidth, alpha, **options):
r"""
``region_plot`` takes a boolean function of two variables, `f(x,y)`
and plots the region where f is True over the specified
``xrange`` and ``yrange`` as demonstrated below.
``region_plot(f, (xmin, xmax), (ymin, ymax), ...)``
INPUT:
- ``f`` -- a boolean function or a list of boolean functions of two variables
- ``(xmin, xmax)`` -- 2-tuple, the range of ``x`` values OR 3-tuple
``(x,xmin,xmax)``
- ``(ymin, ymax)`` -- 2-tuple, the range of ``y`` values OR 3-tuple
``(y,ymin,ymax)``
- ``plot_points`` -- integer (default: 100); number of points to plot
in each direction of the grid
- ``incol`` -- a color (default: ``'blue'``), the color inside the region
- ``outcol`` -- a color (default: ``None``), the color of the outside
of the region
If any of these options are specified, the border will be shown as indicated,
otherwise it is only implicit (with color ``incol``) as the border of the
inside of the region.
- ``bordercol`` -- a color (default: ``None``), the color of the border
(``'black'`` if ``borderwidth`` or ``borderstyle`` is specified but not ``bordercol``)
- ``borderstyle`` -- string (default: 'solid'), one of ``'solid'``,
``'dashed'``, ``'dotted'``, ``'dashdot'``, respectively ``'-'``,
``'--'``, ``':'``, ``'-.'``.
- ``borderwidth`` -- integer (default: None), the width of the border in pixels
- ``alpha`` -- (default: 1) How transparent the fill is. A number between 0 and 1.
- ``legend_label`` -- the label for this item in the legend
- ``base`` - (default: 10) the base of the logarithm if
a logarithmic scale is set. This must be greater than 1. The base
can be also given as a list or tuple ``(basex, basey)``.
``basex`` sets the base of the logarithm along the horizontal
axis and ``basey`` sets the base along the vertical axis.
- ``scale`` -- (default: ``"linear"``) string. The scale of the axes.
Possible values are ``"linear"``, ``"loglog"``, ``"semilogx"``,
``"semilogy"``.
The scale can be also be given as single argument that is a list
or tuple ``(scale, base)`` or ``(scale, basex, basey)``.
The ``"loglog"`` scale sets both the horizontal and vertical axes to
logarithmic scale. The ``"semilogx"`` scale sets the horizontal axis
to logarithmic scale. The ``"semilogy"`` scale sets the vertical axis
to logarithmic scale. The ``"linear"`` scale is the default value
when :class:`~sage.plot.graphics.Graphics` is initialized.
EXAMPLES:
Here we plot a simple function of two variables::
sage: x,y = var('x,y')
sage: region_plot(cos(x^2+y^2) <= 0, (x, -3, 3), (y, -3, 3))
Graphics object consisting of 1 graphics primitive
Here we play with the colors::
sage: region_plot(x^2+y^3 < 2, (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray')
Graphics object consisting of 2 graphics primitives
An even more complicated plot, with dashed borders::
sage: region_plot(sin(x)*sin(y) >= 1/4, (x,-10,10), (y,-10,10), incol='yellow', bordercol='black', borderstyle='dashed', plot_points=250)
Graphics object consisting of 2 graphics primitives
A disk centered at the origin::
sage: region_plot(x^2+y^2<1, (x,-1,1), (y,-1,1))
Graphics object consisting of 1 graphics primitive
A plot with more than one condition (all conditions must be true for the statement to be true)::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2))
Graphics object consisting of 1 graphics primitive
Since it doesn't look very good, let's increase ``plot_points``::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2), plot_points=400)
Graphics object consisting of 1 graphics primitive
To get plots where only one condition needs to be true, use a function.
Using lambda functions, we definitely need the extra ``plot_points``::
sage: region_plot(lambda x,y: x^2+y^2<1 or x<y, (x,-2,2), (y,-2,2), plot_points=400)
Graphics object consisting of 1 graphics primitive
The first quadrant of the unit circle::
sage: region_plot([y>0, x>0, x^2+y^2<1], (x,-1.1, 1.1), (y,-1.1, 1.1), plot_points = 400)
Graphics object consisting of 1 graphics primitive
Here is another plot, with a huge border::
sage: region_plot(x*(x-1)*(x+1)+y^2<0, (x, -3, 2), (y, -3, 3), incol='lightblue', bordercol='gray', borderwidth=10, plot_points=50)
Graphics object consisting of 2 graphics primitives
If we want to keep only the region where x is positive::
sage: region_plot([x*(x-1)*(x+1)+y^2<0, x>-1], (x, -3, 2), (y, -3, 3), incol='lightblue', plot_points=50)
Graphics object consisting of 1 graphics primitive
Here we have a cut circle::
sage: region_plot([x^2+y^2<4, x>-1], (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray', plot_points=200)
Graphics object consisting of 2 graphics primitives
The first variable range corresponds to the horizontal axis and
the second variable range corresponds to the vertical axis::
sage: s,t=var('s,t')
sage: region_plot(s>0,(t,-2,2),(s,-2,2))
Graphics object consisting of 1 graphics primitive
::
sage: region_plot(s>0,(s,-2,2),(t,-2,2))
Graphics object consisting of 1 graphics primitive
An example of a region plot in 'loglog' scale::
sage: region_plot(x^2+y^2<100, (x,1,10), (y,1,10), scale='loglog')
Graphics object consisting of 1 graphics primitive
TESTS:
To check that :trac:`16907` is fixed::
sage: x, y = var('x, y')
sage: disc1 = region_plot(x^2+y^2 < 1, (x, -1, 1), (y, -1, 1), alpha=0.5)
sage: disc2 = region_plot((x-0.7)^2+(y-0.7)^2 < 0.5, (x, -2, 2), (y, -2, 2), incol='red', alpha=0.5)
sage: disc1 + disc2
Graphics object consisting of 2 graphics primitives
To check that :trac:`18286` is fixed::
sage: x, y = var('x, y')
sage: region_plot([x == 0], (x, -1, 1), (y, -1, 1))
Graphics object consisting of 1 graphics primitive
sage: region_plot([x^2+y^2==1, x<y], (x, -1, 1), (y, -1, 1))
Graphics object consisting of 1 graphics primitive
"""
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
from sage.symbolic.expression import is_Expression
from warnings import warn
import numpy
if not isinstance(f, (list, tuple)):
f = [f]
feqs = [
equify(g) for g in f if is_Expression(g)
and g.operator() is operator.eq and not equify(g).is_zero()
]
f = [
equify(g) for g in f
if not (is_Expression(g) and g.operator() is operator.eq)
]
neqs = len(feqs)
if neqs > 1:
warn(
"There are at least 2 equations; If the region is degenerated to points, plotting might show nothing."
)
feqs = [sum([fn**2 for fn in feqs])]
neqs = 1
if neqs and not bordercol:
bordercol = incol
if not f:
return implicit_plot(feqs[0], xrange, yrange, plot_points=plot_points, fill=False, \
linewidth=borderwidth, linestyle=borderstyle, color=bordercol, **options)
f_all, ranges = setup_for_eval_on_grid(feqs + f, [xrange, yrange],
plot_points)
xrange, yrange = [r[:2] for r in ranges]
xy_data_arrays = numpy.asarray(
[[[func(x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)]
for func in f_all[neqs::]],
dtype=float)
xy_data_array = numpy.abs(xy_data_arrays.prod(axis=0))
# Now we need to set entries to negative iff all
# functions were negative at that point.
neg_indices = (xy_data_arrays < 0).all(axis=0)
xy_data_array[neg_indices] = -xy_data_array[neg_indices]
from matplotlib.colors import ListedColormap
incol = rgbcolor(incol)
if outcol:
outcol = rgbcolor(outcol)
cmap = ListedColormap([incol, outcol])
cmap.set_over(outcol, alpha=alpha)
else:
outcol = rgbcolor('white')
cmap = ListedColormap([incol, outcol])
cmap.set_over(outcol, alpha=0)
cmap.set_under(incol, alpha=alpha)
g = Graphics()
# Reset aspect_ratio to 'automatic' in case scale is 'semilog[xy]'.
# Otherwise matplotlib complains.
scale = options.get('scale', None)
if isinstance(scale, (list, tuple)):
scale = scale[0]
if scale == 'semilogy' or scale == 'semilogx':
options['aspect_ratio'] = 'automatic'
g._set_extra_kwds(
Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax']))
if neqs == 0:
g.add_primitive(
ContourPlot(
xy_data_array, xrange, yrange,
dict(contours=[-1e-20, 0, 1e-20],
cmap=cmap,
fill=True,
**options)))
else:
mask = numpy.asarray([[elt > 0 for elt in rows]
for rows in xy_data_array],
dtype=bool)
xy_data_array = numpy.asarray([[
f_all[0](x, y) for x in xsrange(*ranges[0], include_endpoint=True)
] for y in xsrange(*ranges[1], include_endpoint=True)],
dtype=float)
xy_data_array[mask] = None
if bordercol or borderstyle or borderwidth:
cmap = [rgbcolor(bordercol)] if bordercol else ['black']
linestyles = [borderstyle] if borderstyle else None
linewidths = [borderwidth] if borderwidth else None
g.add_primitive(
ContourPlot(
xy_data_array, xrange, yrange,
dict(linestyles=linestyles,
linewidths=linewidths,
contours=[0],
cmap=[bordercol],
fill=False,
**options)))
return g
def BinaryQF_reduced_representatives(D, primitive_only=False):
r"""
Returns a list of inequivalent reduced representatives for the
equivalence classes of positive definite binary forms of
discriminant D.
INPUT:
- `D` -- (integer) A negative discriminant.
- ``primitive_only`` -- (bool, default False) flag controlling whether only
primitive forms are included.
OUTPUT:
(list) A lexicographically-ordered list of inequivalent reduced
representatives for the equivalence classes of positive definite binary
forms of discriminant `D`. If ``primitive_only`` is ``True`` then
imprimitive forms (which only exist when `D` is not fundamental) are
omitted; otherwise they are included.
EXAMPLES::
sage: BinaryQF_reduced_representatives(-4)
[x^2 + y^2]
sage: BinaryQF_reduced_representatives(-163)
[x^2 + x*y + 41*y^2]
sage: BinaryQF_reduced_representatives(-12)
[x^2 + 3*y^2, 2*x^2 + 2*x*y + 2*y^2]
sage: BinaryQF_reduced_representatives(-16)
[x^2 + 4*y^2, 2*x^2 + 2*y^2]
sage: BinaryQF_reduced_representatives(-63)
[x^2 + x*y + 16*y^2, 2*x^2 - x*y + 8*y^2, 2*x^2 + x*y + 8*y^2, 3*x^2 + 3*x*y + 6*y^2, 4*x^2 + x*y + 4*y^2]
The number of inequivalent reduced binary forms with a fixed negative
fundamental discriminant D is the class number of the quadratic field
`Q(\sqrt{D})`::
sage: len(BinaryQF_reduced_representatives(-13*4))
2
sage: QuadraticField(-13*4, 'a').class_number()
2
sage: p=next_prime(2^20); p
1048583
sage: len(BinaryQF_reduced_representatives(-p))
689
sage: QuadraticField(-p, 'a').class_number()
689
sage: BinaryQF_reduced_representatives(-23*9)
[x^2 + x*y + 52*y^2,
2*x^2 - x*y + 26*y^2,
2*x^2 + x*y + 26*y^2,
3*x^2 + 3*x*y + 18*y^2,
4*x^2 - x*y + 13*y^2,
4*x^2 + x*y + 13*y^2,
6*x^2 - 3*x*y + 9*y^2,
6*x^2 + 3*x*y + 9*y^2,
8*x^2 + 7*x*y + 8*y^2]
sage: BinaryQF_reduced_representatives(-23*9, primitive_only=True)
[x^2 + x*y + 52*y^2,
2*x^2 - x*y + 26*y^2,
2*x^2 + x*y + 26*y^2,
4*x^2 - x*y + 13*y^2,
4*x^2 + x*y + 13*y^2,
8*x^2 + 7*x*y + 8*y^2]
TESTS::
sage: BinaryQF_reduced_representatives(5)
Traceback (most recent call last):
...
ValueError: discriminant must be negative and congruent to 0 or 1 modulo 4
"""
D = ZZ(D)
if not ( D < 0 and (D % 4 in [0,1])):
raise ValueError("discriminant must be negative and congruent to 0 or 1 modulo 4")
# For a fundamental discriminant all forms are primitive so we need not check:
if primitive_only:
primitive_only = not is_fundamental_discriminant(D)
form_list = []
from sage.arith.srange import xsrange
# Only iterate over positive a and over b of the same
# parity as D such that 4a^2 + D <= b^2 <= a^2
for a in xsrange(1,1+((-D)//3).isqrt()):
a4 = 4*a
s = D + a*a4
w = 1+(s-1).isqrt() if s > 0 else 0
if w%2 != D%2: w += 1
for b in xsrange(w,a+1,2):
t = b*b-D
if t % a4 == 0:
c = t // a4
if (not primitive_only) or gcd([a,b,c])==1:
if b>0 and a>b and c>a:
form_list.append(BinaryQF([a,-b,c]))
form_list.append(BinaryQF([a,b,c]))
form_list.sort()
return form_list
def streamline_plot(f_g, xrange, yrange, **options):
r"""
Return a streamline plot in a vector field.
``streamline_plot`` can take either one or two functions. Consider
two variables `x` and `y`.
If given two functions `(f(x,y), g(x,y))`, then this function plots
streamlines in the vector field over the specified ranges with ``xrange``
being of `x`, denoted by ``xvar`` below, between ``xmin`` and ``xmax``,
and ``yrange`` similarly (see below). ::
streamline_plot((f, g), (xvar, xmin, xmax), (yvar, ymin, ymax))
Similarly, if given one function `f(x, y)`, then this function plots
streamlines in the slope field `dy/dx = f(x,y)` over the specified
ranges as given above.
PLOT OPTIONS:
- ``plot_points`` -- (default: 200) the minimal number of plot points
- ``density`` -- float (default: 1.); controls the closeness of
streamlines
- ``start_points`` -- (optional) list of coordinates of starting
points for the streamlines; coordinate pairs can be tuples or lists
EXAMPLES:
Plot some vector fields involving `\sin` and `\cos`::
sage: x, y = var('x y')
sage: streamline_plot((sin(x), cos(y)), (x,-3,3), (y,-3,3))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = streamline_plot((sin(x), cos(y)), (x,-3,3), (y,-3,3))
sphinx_plot(g)
::
sage: streamline_plot((y, (cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = streamline_plot((y, (cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi))
sphinx_plot(g)
We increase the density of the plot::
sage: streamline_plot((y, (cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi), density=2)
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = streamline_plot((y, (cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi), density=2)
sphinx_plot(g)
We ignore function values that are infinite or NaN::
sage: x, y = var('x y')
sage: streamline_plot((-x/sqrt(x^2+y^2), -y/sqrt(x^2+y^2)), (x,-10,10), (y,-10,10))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = streamline_plot((-x/sqrt(x**2+y**2), -y/sqrt(x**2+y**2)), (x,-10,10), (y,-10,10))
sphinx_plot(g)
Extra options will get passed on to :func:`show()`, as long as they
are valid::
sage: streamline_plot((x, y), (x,-2,2), (y,-2,2), xmax=10)
Graphics object consisting of 1 graphics primitive
sage: streamline_plot((x, y), (x,-2,2), (y,-2,2)).show(xmax=10) # These are equivalent
.. PLOT::
x, y = var('x y')
g = streamline_plot((x, y), (x,-2,2), (y,-2,2), xmax=10)
sphinx_plot(g)
We can also construct streamlines in a slope field::
sage: x, y = var('x y')
sage: streamline_plot((x + y) / sqrt(x^2 + y^2), (x,-3,3), (y,-3,3))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = streamline_plot((x + y) / sqrt(x**2 + y**2), (x,-3,3), (y,-3,3))
sphinx_plot(g)
We choose some particular points the streamlines pass through::
sage: pts = [[1, 1], [-2, 2], [1, -3/2]]
sage: g = streamline_plot((x + y) / sqrt(x^2 + y^2), (x,-3,3), (y,-3,3), start_points=pts)
sage: g += point(pts, color='red')
sage: g
Graphics object consisting of 2 graphics primitives
.. PLOT::
x, y = var('x y')
pts = [[1, 1], [-2, 2], [1, -3/2]]
g = streamline_plot((x + y) / sqrt(x**2 + y**2), (x,-3,3), (y,-3,3), start_points=pts)
g += point(pts, color='red')
sphinx_plot(g)
.. NOTE::
Streamlines currently pass close to ``start_points`` but do
not necessarily pass directly through them. That is part of
the behavior of matplotlib, not an error on your part.
"""
# Parse the function input
if isinstance(f_g, (list, tuple)):
(f, g) = f_g
else:
from sage.functions.all import sqrt
from inspect import isfunction
if isfunction(f_g):
f = lambda x, y: 1 / sqrt(f_g(x, y)**2 + 1)
g = lambda x, y: f_g(x, y) * f(x, y)
else:
f = 1 / sqrt(f_g**2 + 1)
g = f_g * f
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
z, ranges = setup_for_eval_on_grid([f, g], [xrange, yrange],
options['plot_points'])
f, g = z
# The density values must be floats
if isinstance(options['density'], (list, tuple)):
options['density'] = [float(x) for x in options['density']]
else:
options['density'] = float(options['density'])
xpos_array, ypos_array, xvec_array, yvec_array = [], [], [], []
for x in xsrange(*ranges[0], include_endpoint=True):
xpos_array.append(x)
for y in xsrange(*ranges[1], include_endpoint=True):
ypos_array.append(y)
xvec_row, yvec_row = [], []
for x in xsrange(*ranges[0], include_endpoint=True):
xvec_row.append(f(x, y))
yvec_row.append(g(x, y))
xvec_array.append(xvec_row)
yvec_array.append(yvec_row)
import numpy
xpos_array = numpy.array(xpos_array, dtype=float)
ypos_array = numpy.array(ypos_array, dtype=float)
xvec_array = numpy.ma.masked_invalid(numpy.array(xvec_array, dtype=float))
yvec_array = numpy.ma.masked_invalid(numpy.array(yvec_array, dtype=float))
if 'start_points' in options:
xstart_array, ystart_array = [], []
for point in options['start_points']:
xstart_array.append(point[0])
ystart_array.append(point[1])
options['start_points'] = numpy.array([xstart_array, ystart_array]).T
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options))
g.add_primitive(
StreamlinePlot(xpos_array, ypos_array, xvec_array, yvec_array,
options))
return g
def density_plot(f, xrange, yrange, **options):
r"""
``density_plot`` takes a function of two variables, `f(x,y)`
and plots the height of the function over the specified
``xrange`` and ``yrange`` as demonstrated below.
``density_plot(f, (xmin,xmax), (ymin,ymax), ...)``
INPUT:
- ``f`` -- a function of two variables
- ``(xmin,xmax)`` -- 2-tuple, the range of ``x`` values OR 3-tuple
``(x,xmin,xmax)``
- ``(ymin,ymax)`` -- 2-tuple, the range of ``y`` values OR 3-tuple
``(y,ymin,ymax)``
The following inputs must all be passed in as named parameters:
- ``plot_points`` -- integer (default: 25); number of points to plot
in each direction of the grid
- ``cmap`` -- a colormap (default: ``'gray'``), the name of
a predefined colormap, a list of colors or an instance of a matplotlib
Colormap. Type: ``import matplotlib.cm; matplotlib.cm.datad.keys()``
for available colormap names.
- ``interpolation`` -- string (default: ``'catrom'``), the interpolation
method to use: ``'bilinear'``, ``'bicubic'``, ``'spline16'``,
``'spline36'``, ``'quadric'``, ``'gaussian'``, ``'sinc'``,
``'bessel'``, ``'mitchell'``, ``'lanczos'``, ``'catrom'``,
``'hermite'``, ``'hanning'``, ``'hamming'``, ``'kaiser'``
EXAMPLES:
Here we plot a simple function of two variables. Note that
since the input function is an expression, we need to explicitly
declare the variables in 3-tuples for the range::
sage: x,y = var('x,y')
sage: density_plot(sin(x) * sin(y), (x,-2,2), (y,-2,2))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
g = density_plot(sin(x) * sin(y), (x,-2,2), (y,-2,2))
sphinx_plot(g)
Here we change the ranges and add some options; note that here
``f`` is callable (has variables declared), so we can use 2-tuple ranges::
sage: x,y = var('x,y')
sage: f(x,y) = x^2 * cos(x*y)
sage: density_plot(f, (x,-10,5), (y,-5,5), interpolation='sinc', plot_points=100)
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
def f(x,y): return x**2 * cos(x*y)
g = density_plot(f, (x,-10,5), (y,-5,5), interpolation='sinc', plot_points=100)
sphinx_plot(g)
An even more complicated plot::
sage: x,y = var('x,y')
sage: density_plot(sin(x^2+y^2) * cos(x) * sin(y), (x,-4,4), (y,-4,4), cmap='jet', plot_points=100)
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
g = density_plot(sin(x**2 + y**2)*cos(x)*sin(y), (x,-4,4), (y,-4,4), cmap='jet', plot_points=100)
sphinx_plot(g)
This should show a "spotlight" right on the origin::
sage: x,y = var('x,y')
sage: density_plot(1/(x^10 + y^10), (x,-10,10), (y,-10,10))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
g = density_plot(1/(x**10 + y**10), (x,-10,10), (y,-10,10))
sphinx_plot(g)
Some elliptic curves, but with symbolic endpoints. In the first
example, the plot is rotated 90 degrees because we switch the
variables `x`, `y`::
sage: density_plot(y^2 + 1 - x^3 - x, (y,-pi,pi), (x,-pi,pi))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
g = density_plot(y**2 + 1 - x**3 - x, (y,-pi,pi), (x,-pi,pi))
sphinx_plot(g)
::
sage: density_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
g = density_plot(y**2 + 1 - x**3 - x, (x,-pi,pi), (y,-pi,pi))
sphinx_plot(g)
Extra options will get passed on to show(), as long as they are valid::
sage: density_plot(log(x) + log(y), (x,1,10), (y,1,10), dpi=20)
Graphics object consisting of 1 graphics primitive
.. PLOT::
x,y = var('x,y')
g = density_plot(log(x) + log(y), (x,1,10), (y,1,10), dpi=20)
sphinx_plot(g)
::
sage: density_plot(log(x) + log(y), (x,1,10), (y,1,10)).show(dpi=20) # These are equivalent
TESTS:
Check that :trac:`15315` is fixed, i.e., density_plot respects the
``aspect_ratio`` parameter. Without the fix, it looks like a thin line
of width a few mm. With the fix it should look like a nice fat layered
image::
sage: density_plot((x*y)^(1/2), (x,0,3), (y,0,500), aspect_ratio=.01)
Graphics object consisting of 1 graphics primitive
Default ``aspect_ratio`` is ``"automatic"``, and that should work too::
sage: density_plot((x*y)^(1/2), (x,0,3), (y,0,500))
Graphics object consisting of 1 graphics primitive
"""
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid([f], [xrange, yrange], options['plot_points'])
g = g[0]
xrange, yrange = [r[:2] for r in ranges]
xy_data_array = [[g(x,y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)]
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=['xmin','xmax']))
g.add_primitive(DensityPlot(xy_data_array, xrange, yrange, options))
return g
def _render_on_subplot(self, subplot):
"""
TESTS:
A somewhat random plot, but fun to look at::
sage: x,y = var('x,y')
sage: contour_plot(x^2-y^3+10*sin(x*y), (x, -4, 4), (y, -4, 4),plot_points=121,cmap='hsv')
Graphics object consisting of 1 graphics primitive
"""
from sage.rings.integer import Integer
options = self.options()
fill = options['fill']
contours = options['contours']
if 'cmap' in options:
cmap = get_cmap(options['cmap'])
elif fill or contours is None:
cmap = get_cmap('gray')
else:
if isinstance(contours, (int, Integer)):
cmap = get_cmap([(i,i,i) for i in xsrange(0,1,1/contours)])
else:
l = Integer(len(contours))
cmap = get_cmap([(i,i,i) for i in xsrange(0,1,1/l)])
x0,x1 = float(self.xrange[0]), float(self.xrange[1])
y0,y1 = float(self.yrange[0]), float(self.yrange[1])
if isinstance(contours, (int, Integer)):
contours = int(contours)
CSF=None
if fill:
if contours is None:
CSF=subplot.contourf(self.xy_data_array, cmap=cmap, extent=(x0,x1,y0,y1), label=options['legend_label'])
else:
CSF=subplot.contourf(self.xy_data_array, contours, cmap=cmap, extent=(x0,x1,y0,y1),extend='both', label=options['legend_label'])
linewidths = options.get('linewidths',None)
if isinstance(linewidths, (int, Integer)):
linewidths = int(linewidths)
elif isinstance(linewidths, (list, tuple)):
linewidths = tuple(int(x) for x in linewidths)
from sage.plot.misc import get_matplotlib_linestyle
linestyles = options.get('linestyles', None)
if isinstance(linestyles, (list, tuple)):
linestyles = [get_matplotlib_linestyle(l, 'long') for l in linestyles]
else:
linestyles = get_matplotlib_linestyle(linestyles, 'long')
if contours is None:
CS = subplot.contour(self.xy_data_array, cmap=cmap, extent=(x0,x1,y0,y1),
linewidths=linewidths, linestyles=linestyles, label=options['legend_label'])
else:
CS = subplot.contour(self.xy_data_array, contours, cmap=cmap, extent=(x0,x1,y0,y1),
linewidths=linewidths, linestyles=linestyles, label=options['legend_label'])
if options.get('labels', False):
label_options = options['label_options']
label_options['fontsize'] = int(label_options['fontsize'])
if fill and label_options is None:
label_options['inline']=False
subplot.clabel(CS, **label_options)
if options.get('colorbar', False):
colorbar_options = options['colorbar_options']
from matplotlib import colorbar
cax,kwds=colorbar.make_axes_gridspec(subplot,**colorbar_options)
if CSF is None:
cb=colorbar.Colorbar(cax,CS, **kwds)
else:
cb=colorbar.Colorbar(cax,CSF, **kwds)
cb.add_lines(CS)
def bsgs(a, b, bounds, operation='*', identity=None, inverse=None, op=None):
r"""
Totally generic discrete baby-step giant-step function.
Solves `na=b` (or `a^n=b`) with `lb\le n\le ub` where ``bounds==(lb,ub)``,
raising an error if no such `n` exists.
`a` and `b` must be elements of some group with given identity,
inverse of ``x`` given by ``inverse(x)``, and group operation on
``x``, ``y`` by ``op(x,y)``.
If operation is '*' or '+' then the other
arguments are provided automatically; otherwise they must be
provided by the caller.
INPUT:
- ``a`` - group element
- ``b`` - group element
- ``bounds`` - a 2-tuple of integers ``(lower,upper)`` with ``0<=lower<=upper``
- ``operation`` - string: '*', '+', 'other'
- ``identity`` - the identity element of the group
- ``inverse()`` - function of 1 argument ``x`` returning inverse of ``x``
- ``op()`` - function of 2 arguments ``x``, ``y`` returning ``x*y`` in group
OUTPUT:
An integer `n` such that `a^n = b` (or `na = b`). If no
such `n` exists, this function raises a ValueError exception.
NOTE: This is a generalization of discrete logarithm. One
situation where this version is useful is to find the order of
an element in a group where we only have bounds on the group
order (see the elliptic curve example below).
ALGORITHM: Baby step giant step. Time and space are soft
`O(\sqrt{n})` where `n` is the difference between upper and lower
bounds.
EXAMPLES::
sage: b = Mod(2,37); a = b^20
sage: bsgs(b, a, (0,36))
20
sage: p=next_prime(10^20)
sage: a=Mod(2,p); b=a^(10^25)
sage: bsgs(a, b, (10^25-10^6,10^25+10^6)) == 10^25
True
sage: K = GF(3^6,'b')
sage: a = K.gen()
sage: b = a^210
sage: bsgs(a, b, (0,K.order()-1))
210
sage: K.<z>=CyclotomicField(230)
sage: w=z^500
sage: bsgs(z,w,(0,229))
40
An additive example in an elliptic curve group::
sage: F.<a> = GF(37^5)
sage: E = EllipticCurve(F, [1,1])
sage: P = E.lift_x(a); P
(a : 28*a^4 + 15*a^3 + 14*a^2 + 7 : 1)
This will return a multiple of the order of P::
sage: bsgs(P,P.parent()(0),Hasse_bounds(F.order()),operation='+')
69327408
AUTHOR:
- John Cremona (2008-03-15)
"""
Z = integer_ring.ZZ
from operator import inv, mul, neg, add
if operation in multiplication_names:
identity = a.parent()(1)
inverse = inv
op = mul
elif operation in addition_names:
identity = a.parent()(0)
inverse = neg
op = add
else:
if identity is None or inverse is None or op is None:
raise ValueError("identity, inverse and operation must be given")
lb, ub = bounds
if lb<0 or ub<lb:
raise ValueError("bsgs() requires 0<=lb<=ub")
if a.is_zero() and not b.is_zero():
raise ValueError("No solution in bsgs()")
ran = 1 + ub - lb # the length of the interval
c = op(inverse(b),multiple(a,lb,operation=operation))
if ran < 30: # use simple search for small ranges
i = lb
d = c
# for i,d in multiples(a,ran,c,indexed=True,operation=operation):
for i0 in range(ran):
i = lb + i0
if identity == d: # identity == b^(-1)*a^i, so return i
return Z(i)
d = op(a,d)
raise ValueError("No solution in bsgs()")
m = ran.isqrt()+1 # we need sqrt(ran) rounded up
table = dict() # will hold pairs (a^(lb+i),lb+i) for i in range(m)
d=c
for i0 in xsrange(m):
i = lb + i0
if identity==d: # identity == b^(-1)*a^i, so return i
return Z(i)
table[d] = i
d=op(d,a)
c = op(c,inverse(d)) # this is now a**(-m)
d=identity
for i in xsrange(m):
j = table.get(d)
if j is not None: # then d == b*a**(-i*m) == a**j
return Z(i*m + j)
d=op(c,d)
raise ValueError("Log of %s to the base %s does not exist in %s."%(b,a,bounds))
def BinaryQF_reduced_representatives(D, primitive_only=False):
r"""
Returns a list of inequivalent reduced representatives for the
equivalence classes of positive definite binary forms of
discriminant D.
INPUT:
- `D` -- (integer) A negative discriminant.
- ``primitive_only`` -- (bool, default False) flag controlling whether only
primitive forms are included.
OUTPUT:
(list) A lexicographically-ordered list of inequivalent reduced
representatives for the equivalence classes of positive definite binary
forms of discriminant `D`. If ``primitive_only`` is ``True`` then
imprimitive forms (which only exist when `D` is not fundamental) are
omitted; otherwise they are included.
EXAMPLES::
sage: BinaryQF_reduced_representatives(-4)
[x^2 + y^2]
sage: BinaryQF_reduced_representatives(-163)
[x^2 + x*y + 41*y^2]
sage: BinaryQF_reduced_representatives(-12)
[x^2 + 3*y^2, 2*x^2 + 2*x*y + 2*y^2]
sage: BinaryQF_reduced_representatives(-16)
[x^2 + 4*y^2, 2*x^2 + 2*y^2]
sage: BinaryQF_reduced_representatives(-63)
[x^2 + x*y + 16*y^2, 2*x^2 - x*y + 8*y^2, 2*x^2 + x*y + 8*y^2, 3*x^2 + 3*x*y + 6*y^2, 4*x^2 + x*y + 4*y^2]
The number of inequivalent reduced binary forms with a fixed negative
fundamental discriminant D is the class number of the quadratic field
`Q(\sqrt{D})`::
sage: len(BinaryQF_reduced_representatives(-13*4))
2
sage: QuadraticField(-13*4, 'a').class_number()
2
sage: p=next_prime(2^20); p
1048583
sage: len(BinaryQF_reduced_representatives(-p))
689
sage: QuadraticField(-p, 'a').class_number()
689
sage: BinaryQF_reduced_representatives(-23*9)
[x^2 + x*y + 52*y^2,
2*x^2 - x*y + 26*y^2,
2*x^2 + x*y + 26*y^2,
3*x^2 + 3*x*y + 18*y^2,
4*x^2 - x*y + 13*y^2,
4*x^2 + x*y + 13*y^2,
6*x^2 - 3*x*y + 9*y^2,
6*x^2 + 3*x*y + 9*y^2,
8*x^2 + 7*x*y + 8*y^2]
sage: BinaryQF_reduced_representatives(-23*9, primitive_only=True)
[x^2 + x*y + 52*y^2,
2*x^2 - x*y + 26*y^2,
2*x^2 + x*y + 26*y^2,
4*x^2 - x*y + 13*y^2,
4*x^2 + x*y + 13*y^2,
8*x^2 + 7*x*y + 8*y^2]
TESTS::
sage: BinaryQF_reduced_representatives(5)
Traceback (most recent call last):
...
ValueError: discriminant must be negative and congruent to 0 or 1 modulo 4
"""
D = ZZ(D)
if not ( D < 0 and (D % 4 in [0,1])):
raise ValueError("discriminant must be negative and congruent to 0 or 1 modulo 4")
# For a fundamental discriminant all forms are primitive so we need not check:
if primitive_only:
primitive_only = not is_fundamental_discriminant(D)
form_list = []
from sage.arith.srange import xsrange
# Only iterate over positive a and over b of the same
# parity as D such that 4a^2 + D <= b^2 <= a^2
for a in xsrange(1,1+((-D)//3).isqrt()):
a4 = 4*a
s = D + a*a4
w = 1+(s-1).isqrt() if s > 0 else 0
if w%2 != D%2: w += 1
for b in xsrange(w,a+1,2):
t = b*b-D
if t % a4 == 0:
c = t // a4
if (not primitive_only) or gcd([a,b,c])==1:
if b>0 and a>b and c>a:
form_list.append(BinaryQF([a,-b,c]))
form_list.append(BinaryQF([a,b,c]))
form_list.sort()
return form_list
def bsgs(a, b, bounds, operation='*', identity=None, inverse=None, op=None):
r"""
Totally generic discrete baby-step giant-step function.
Solves `na=b` (or `a^n=b`) with `lb\le n\le ub` where ``bounds==(lb,ub)``,
raising an error if no such `n` exists.
`a` and `b` must be elements of some group with given identity,
inverse of ``x`` given by ``inverse(x)``, and group operation on
``x``, ``y`` by ``op(x,y)``.
If operation is '*' or '+' then the other
arguments are provided automatically; otherwise they must be
provided by the caller.
INPUT:
- ``a`` - group element
- ``b`` - group element
- ``bounds`` - a 2-tuple of integers ``(lower,upper)`` with ``0<=lower<=upper``
- ``operation`` - string: '*', '+', 'other'
- ``identity`` - the identity element of the group
- ``inverse()`` - function of 1 argument ``x`` returning inverse of ``x``
- ``op()`` - function of 2 arguments ``x``, ``y`` returning ``x*y`` in group
OUTPUT:
An integer `n` such that `a^n = b` (or `na = b`). If no
such `n` exists, this function raises a ValueError exception.
NOTE: This is a generalization of discrete logarithm. One
situation where this version is useful is to find the order of
an element in a group where we only have bounds on the group
order (see the elliptic curve example below).
ALGORITHM: Baby step giant step. Time and space are soft
`O(\sqrt{n})` where `n` is the difference between upper and lower
bounds.
EXAMPLES::
sage: from sage.groups.generic import bsgs
sage: b = Mod(2,37); a = b^20
sage: bsgs(b, a, (0,36))
20
sage: p=next_prime(10^20)
sage: a=Mod(2,p); b=a^(10^25)
sage: bsgs(a, b, (10^25-10^6,10^25+10^6)) == 10^25
True
sage: K = GF(3^6,'b')
sage: a = K.gen()
sage: b = a^210
sage: bsgs(a, b, (0,K.order()-1))
210
sage: K.<z>=CyclotomicField(230)
sage: w=z^500
sage: bsgs(z,w,(0,229))
40
An additive example in an elliptic curve group::
sage: F.<a> = GF(37^5)
sage: E = EllipticCurve(F, [1,1])
sage: P = E.lift_x(a); P
(a : 28*a^4 + 15*a^3 + 14*a^2 + 7 : 1)
This will return a multiple of the order of P::
sage: bsgs(P,P.parent()(0),Hasse_bounds(F.order()),operation='+')
69327408
AUTHOR:
- John Cremona (2008-03-15)
"""
Z = integer_ring.ZZ
from operator import inv, mul, neg, add
if operation in multiplication_names:
identity = a.parent()(1)
inverse = inv
op = mul
elif operation in addition_names:
identity = a.parent()(0)
inverse = neg
op = add
else:
if identity is None or inverse is None or op is None:
raise ValueError("identity, inverse and operation must be given")
lb, ub = bounds
if lb < 0 or ub < lb:
raise ValueError("bsgs() requires 0<=lb<=ub")
if a.is_zero() and not b.is_zero():
raise ValueError("No solution in bsgs()")
ran = 1 + ub - lb # the length of the interval
c = op(inverse(b), multiple(a, lb, operation=operation))
if ran < 30: # use simple search for small ranges
i = lb
d = c
# for i,d in multiples(a,ran,c,indexed=True,operation=operation):
for i0 in range(ran):
i = lb + i0
if identity == d: # identity == b^(-1)*a^i, so return i
return Z(i)
d = op(a, d)
raise ValueError("No solution in bsgs()")
m = ran.isqrt() + 1 # we need sqrt(ran) rounded up
table = dict() # will hold pairs (a^(lb+i),lb+i) for i in range(m)
d = c
for i0 in xsrange(m):
i = lb + i0
if identity == d: # identity == b^(-1)*a^i, so return i
return Z(i)
table[d] = i
d = op(d, a)
c = op(c, inverse(d)) # this is now a**(-m)
d = identity
for i in xsrange(m):
j = table.get(d)
if j is not None: # then d == b*a**(-i*m) == a**j
return Z(i * m + j)
d = op(c, d)
raise ValueError("Log of %s to the base %s does not exist in %s." %
(b, a, bounds))
def contour_plot(f, xrange, yrange, **options):
r"""
``contour_plot`` takes a function of two variables, `f(x,y)`
and plots contour lines of the function over the specified
``xrange`` and ``yrange`` as demonstrated below.
``contour_plot(f, (xmin, xmax), (ymin, ymax), ...)``
INPUT:
- ``f`` -- a function of two variables
- ``(xmin, xmax)`` -- 2-tuple, the range of ``x`` values OR 3-tuple
``(x,xmin,xmax)``
- ``(ymin, ymax)`` -- 2-tuple, the range of ``y`` values OR 3-tuple
``(y,ymin,ymax)``
The following inputs must all be passed in as named parameters:
- ``plot_points`` -- integer (default: 100); number of points to plot
in each direction of the grid. For old computers, 25 is fine, but
should not be used to verify specific intersection points.
- ``fill`` -- bool (default: ``True``), whether to color in the area
between contour lines
- ``cmap`` -- a colormap (default: ``'gray'``), the name of
a predefined colormap, a list of colors or an instance of a matplotlib
Colormap. Type: ``import matplotlib.cm; matplotlib.cm.datad.keys()``
for available colormap names.
- ``contours`` -- integer or list of numbers (default: ``None``):
If a list of numbers is given, then this specifies the contour levels
to use. If an integer is given, then this many contour lines are
used, but the exact levels are determined automatically. If ``None``
is passed (or the option is not given), then the number of contour
lines is determined automatically, and is usually about 5.
- ``linewidths`` -- integer or list of integer (default: None), if
a single integer all levels will be of the width given,
otherwise the levels will be plotted with the width in the order
given. If the list is shorter than the number of contours, then
the widths will be repeated cyclically.
- ``linestyles`` -- string or list of strings (default: None), the
style of the lines to be plotted, one of: ``"solid"``, ``"dashed"``,
``"dashdot"``, ``"dotted"``, respectively ``"-"``, ``"--"``,
``"-."``, ``":"``. If the list is shorter than the number of
contours, then the styles will be repeated cyclically.
- ``labels`` -- boolean (default: False) Show level labels or not.
The following options are to adjust the style and placement of
labels, they have no effect if no labels are shown.
- ``label_fontsize`` -- integer (default: 9), the font size of the labels.
- ``label_colors`` -- string or sequence of colors (default:
None) If a string, gives the name of a single color with which
to draw all labels. If a sequence, gives the colors of the
labels. A color is a string giving the name of one or a
3-tuple of floats.
- ``label_inline`` -- boolean (default: False if fill is True,
otherwise True), controls whether the underlying contour is
removed or not.
- ``label_inline_spacing`` -- integer (default: 3), When inline,
this is the amount of contour that is removed from each side,
in pixels.
- ``label_fmt`` -- a format string (default: "%1.2f"), this is
used to get the label text from the level. This can also be a
dictionary with the contour levels as keys and corresponding
text string labels as values. It can also be any callable which
returns a string when called with a numeric contour level.
- ``colorbar`` -- boolean (default: False) Show a colorbar or not.
The following options are to adjust the style and placement of
colorbars. They have no effect if a colorbar is not shown.
- ``colorbar_orientation`` -- string (default: 'vertical'),
controls placement of the colorbar, can be either 'vertical'
or 'horizontal'
- ``colorbar_format`` -- a format string, this is used to format
the colorbar labels.
- ``colorbar_spacing`` -- string (default: 'proportional'). If
'proportional', make the contour divisions proportional to
values. If 'uniform', space the colorbar divisions uniformly,
without regard for numeric values.
- ``legend_label`` -- the label for this item in the legend
- ``region`` - (default: None) If region is given, it must be a function
of two variables. Only segments of the surface where region(x,y) returns a
number >0 will be included in the plot.
EXAMPLES:
Here we plot a simple function of two variables. Note that
since the input function is an expression, we need to explicitly
declare the variables in 3-tuples for the range::
sage: x,y = var('x,y')
sage: contour_plot(cos(x^2+y^2), (x, -4, 4), (y, -4, 4))
Graphics object consisting of 1 graphics primitive
Here we change the ranges and add some options::
sage: x,y = var('x,y')
sage: contour_plot((x^2)*cos(x*y), (x, -10, 5), (y, -5, 5), fill=False, plot_points=150)
Graphics object consisting of 1 graphics primitive
An even more complicated plot::
sage: x,y = var('x,y')
sage: contour_plot(sin(x^2 + y^2)*cos(x)*sin(y), (x, -4, 4), (y, -4, 4),plot_points=150)
Graphics object consisting of 1 graphics primitive
Some elliptic curves, but with symbolic endpoints. In the first
example, the plot is rotated 90 degrees because we switch the
variables `x`, `y`::
sage: x,y = var('x,y')
sage: contour_plot(y^2 + 1 - x^3 - x, (y,-pi,pi), (x,-pi,pi))
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
We can play with the contour levels::
sage: x,y = var('x,y')
sage: f(x,y) = x^2 + y^2
sage: contour_plot(f, (-2, 2), (-2, 2))
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (-2, 2), (-2, 2), contours=2, cmap=[(1,0,0), (0,1,0), (0,0,1)])
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (-2, 2), (-2, 2), contours=(0.1, 1.0, 1.2, 1.4), cmap='hsv')
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (-2, 2), (-2, 2), contours=(1.0,), fill=False)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(x-y^2,(x,-5,5),(y,-3,3),contours=[-4,0,1])
Graphics object consisting of 1 graphics primitive
We can change the style of the lines::
sage: contour_plot(f, (-2,2), (-2,2), fill=False, linewidths=10)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (-2,2), (-2,2), fill=False, linestyles='dashdot')
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(x^2-y^2,(x,-3,3),(y,-3,3),contours=[0,1,2,3,4],\
... linewidths=[1,5],linestyles=['solid','dashed'],fill=False)
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(x^2-y^2,(x,-3,3),(y,-3,3),contours=[0,1,2,3,4],\
... linewidths=[1,5],linestyles=['solid','dashed'])
sage: P
Graphics object consisting of 1 graphics primitive
sage: P=contour_plot(x^2-y^2,(x,-3,3),(y,-3,3),contours=[0,1,2,3,4],\
... linewidths=[1,5],linestyles=['-',':'])
sage: P
Graphics object consisting of 1 graphics primitive
We can add labels and play with them::
sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv', labels=True)
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv',\
... labels=True, label_fmt="%1.0f", label_colors='black')
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv',labels=True,\
... contours=[-4,0,4], label_fmt={-4:"low", 0:"medium", 4: "hi"}, label_colors='black')
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv',labels=True,\
... contours=[-4,0,4], label_fmt=lambda x: "$z=%s$"%x, label_colors='black', label_inline=True, \
... label_fontsize=12)
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), \
... fill=False, cmap='hsv', labels=True, label_fontsize=18)
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), \
... fill=False, cmap='hsv', labels=True, label_inline_spacing=1)
sage: P
Graphics object consisting of 1 graphics primitive
::
sage: P= contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), \
... fill=False, cmap='hsv', labels=True, label_inline=False)
sage: P
Graphics object consisting of 1 graphics primitive
We can change the color of the labels if so desired::
sage: contour_plot(f, (-2,2), (-2,2), labels=True, label_colors='red')
Graphics object consisting of 1 graphics primitive
We can add a colorbar as well::
sage: f(x,y)=x^2-y^2
sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True,colorbar_orientation='horizontal')
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4],colorbar=True)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4],colorbar=True,colorbar_spacing='uniform')
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[0,2,3,6],colorbar=True,colorbar_format='%.3f')
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), labels=True,label_colors='red',contours=[0,2,3,6],colorbar=True)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), cmap='winter', contours=20, fill=False, colorbar=True)
Graphics object consisting of 1 graphics primitive
This should plot concentric circles centered at the origin::
sage: x,y = var('x,y')
sage: contour_plot(x^2+y^2-2,(x,-1,1), (y,-1,1))
Graphics object consisting of 1 graphics primitive
Extra options will get passed on to show(), as long as they are valid::
sage: f(x, y) = cos(x) + sin(y)
sage: contour_plot(f, (0, pi), (0, pi), axes=True)
Graphics object consisting of 1 graphics primitive
One can also plot over a reduced region::
sage: contour_plot(x**2-y**2, (x,-2, 2), (y,-2, 2),region=x-y,plot_points=300)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (0, pi), (0, pi)).show(axes=True) # These are equivalent
Note that with ``fill=False`` and grayscale contours, there is the
possibility of confusion between the contours and the axes, so use
``fill=False`` together with ``axes=True`` with caution::
sage: contour_plot(f, (-pi, pi), (-pi, pi), fill=False, axes=True)
Graphics object consisting of 1 graphics primitive
TESTS:
To check that :trac:`5221` is fixed, note that this has three curves, not two::
sage: x,y = var('x,y')
sage: contour_plot(x-y^2,(x,-5,5),(y,-3,3),contours=[-4,-2,0], fill=False)
Graphics object consisting of 1 graphics primitive
"""
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
region = options.pop('region')
ev = [f] if region is None else [f,region]
F, ranges = setup_for_eval_on_grid(ev, [xrange, yrange], options['plot_points'])
g = F[0]
xrange,yrange=[r[:2] for r in ranges]
xy_data_array = [[g(x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)]
if region is not None:
import numpy
xy_data_array = numpy.ma.asarray(xy_data_array,dtype=float)
m = F[1]
mask = numpy.asarray([[m(x, y)<=0 for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)],dtype=bool)
xy_data_array[mask] = numpy.ma.masked
g = Graphics()
# Reset aspect_ratio to 'automatic' in case scale is 'semilog[xy]'.
# Otherwise matplotlib complains.
scale = options.get('scale', None)
if isinstance(scale, (list, tuple)):
scale = scale[0]
if scale == 'semilogy' or scale == 'semilogx':
options['aspect_ratio'] = 'automatic'
g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax']))
g.add_primitive(ContourPlot(xy_data_array, xrange, yrange, options))
return g
def density_plot(f, xrange, yrange, **options):
r"""
``density_plot`` takes a function of two variables, `f(x,y)`
and plots the height of of the function over the specified
``xrange`` and ``yrange`` as demonstrated below.
``density_plot(f, (xmin, xmax), (ymin, ymax), ...)``
INPUT:
- ``f`` -- a function of two variables
- ``(xmin, xmax)`` -- 2-tuple, the range of ``x`` values OR 3-tuple
``(x,xmin,xmax)``
- ``(ymin, ymax)`` -- 2-tuple, the range of ``y`` values OR 3-tuple
``(y,ymin,ymax)``
The following inputs must all be passed in as named parameters:
- ``plot_points`` -- integer (default: 25); number of points to plot
in each direction of the grid
- ``cmap`` -- a colormap (type ``cmap_help()`` for more information).
- ``interpolation`` -- string (default: ``'catrom'``), the interpolation
method to use: ``'bilinear'``, ``'bicubic'``, ``'spline16'``,
``'spline36'``, ``'quadric'``, ``'gaussian'``, ``'sinc'``,
``'bessel'``, ``'mitchell'``, ``'lanczos'``, ``'catrom'``,
``'hermite'``, ``'hanning'``, ``'hamming'``, ``'kaiser'``
EXAMPLES:
Here we plot a simple function of two variables. Note that
since the input function is an expression, we need to explicitly
declare the variables in 3-tuples for the range::
sage: x,y = var('x,y')
sage: density_plot(sin(x)*sin(y), (x, -2, 2), (y, -2, 2))
Graphics object consisting of 1 graphics primitive
Here we change the ranges and add some options; note that here
``f`` is callable (has variables declared), so we can use 2-tuple ranges::
sage: x,y = var('x,y')
sage: f(x,y) = x^2*cos(x*y)
sage: density_plot(f, (x,-10,5), (y, -5,5), interpolation='sinc', plot_points=100)
Graphics object consisting of 1 graphics primitive
An even more complicated plot::
sage: x,y = var('x,y')
sage: density_plot(sin(x^2 + y^2)*cos(x)*sin(y), (x, -4, 4), (y, -4, 4), cmap='jet', plot_points=100)
Graphics object consisting of 1 graphics primitive
This should show a "spotlight" right on the origin::
sage: x,y = var('x,y')
sage: density_plot(1/(x^10+y^10), (x, -10, 10), (y, -10, 10))
Graphics object consisting of 1 graphics primitive
Some elliptic curves, but with symbolic endpoints. In the first
example, the plot is rotated 90 degrees because we switch the
variables `x`, `y`::
sage: density_plot(y^2 + 1 - x^3 - x, (y,-pi,pi), (x,-pi,pi))
Graphics object consisting of 1 graphics primitive
::
sage: density_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
Extra options will get passed on to show(), as long as they are valid::
sage: density_plot(log(x) + log(y), (x, 1, 10), (y, 1, 10), dpi=20)
Graphics object consisting of 1 graphics primitive
::
sage: density_plot(log(x) + log(y), (x, 1, 10), (y, 1, 10)).show(dpi=20) # These are equivalent
TESTS:
Check that :trac:`15315` is fixed, i.e., density_plot respects the
``aspect_ratio`` parameter. Without the fix, it looks like a thin line
of width a few mm. With the fix it should look like a nice fat layered
image::
sage: density_plot((x*y)^(1/2), (x,0,3), (y,0,500), aspect_ratio=.01)
Graphics object consisting of 1 graphics primitive
Default ``aspect_ratio`` is ``"automatic"``, and that should work too::
sage: density_plot((x*y)^(1/2), (x,0,3), (y,0,500))
Graphics object consisting of 1 graphics primitive
"""
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid([f], [xrange, yrange], options['plot_points'])
g = g[0]
xrange,yrange=[r[:2] for r in ranges]
xy_data_array = [[g(x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)]
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax']))
g.add_primitive(DensityPlot(xy_data_array, xrange, yrange, options))
return g
def streamline_plot(f_g, xrange, yrange, **options):
r"""
Return a streamline plot in a vector field.
``streamline_plot`` can take either one or two functions. Consider
two variables `x` and `y`.
If given two functions `(f(x,y), g(x,y))`, then this function plots
streamlines in the vector field over the specified ranges with ``xrange``
being of `x`, denoted by ``xvar`` below, between ``xmin`` and ``xmax``,
and ``yrange`` similarly (see below). ::
streamline_plot((f, g), (xvar, xmin, xmax), (yvar, ymin, ymax))
Similarly, if given one function `f(x, y)`, then this function plots
streamlines in the slope field `dy/dx = f(x,y)` over the specified
ranges as given above.
PLOT OPTIONS:
- ``plot_points`` -- (default: 200) the minimal number of plot points
- ``density`` -- float (default: 1.); controls the closeness of
streamlines
- ``start_points`` -- (optional) list of coordinates of starting
points for the streamlines; coordinate pairs can be tuples or lists
EXAMPLES:
Plot some vector fields involving `\sin` and `\cos`::
sage: x, y = var('x y')
sage: streamline_plot((sin(x), cos(y)), (x,-3,3), (y,-3,3))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = streamline_plot((sin(x), cos(y)), (x,-3,3), (y,-3,3))
sphinx_plot(g)
::
sage: streamline_plot((y, (cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = streamline_plot((y, (cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi))
sphinx_plot(g)
We increase the density of the plot::
sage: streamline_plot((y, (cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi), density=2)
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = streamline_plot((y, (cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi), density=2)
sphinx_plot(g)
We ignore function values that are infinite or NaN::
sage: x, y = var('x y')
sage: streamline_plot((-x/sqrt(x^2+y^2), -y/sqrt(x^2+y^2)), (x,-10,10), (y,-10,10))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = streamline_plot((-x/sqrt(x**2+y**2), -y/sqrt(x**2+y**2)), (x,-10,10), (y,-10,10))
sphinx_plot(g)
Extra options will get passed on to :func:`show()`, as long as they
are valid::
sage: streamline_plot((x, y), (x,-2,2), (y,-2,2), xmax=10)
Graphics object consisting of 1 graphics primitive
sage: streamline_plot((x, y), (x,-2,2), (y,-2,2)).show(xmax=10) # These are equivalent
.. PLOT::
x, y = var('x y')
g = streamline_plot((x, y), (x,-2,2), (y,-2,2), xmax=10)
sphinx_plot(g)
We can also construct streamlines in a slope field::
sage: x, y = var('x y')
sage: streamline_plot((x + y) / sqrt(x^2 + y^2), (x,-3,3), (y,-3,3))
Graphics object consisting of 1 graphics primitive
.. PLOT::
x, y = var('x y')
g = streamline_plot((x + y) / sqrt(x**2 + y**2), (x,-3,3), (y,-3,3))
sphinx_plot(g)
We choose some particular points the streamlines pass through::
sage: pts = [[1, 1], [-2, 2], [1, -3/2]]
sage: g = streamline_plot((x + y) / sqrt(x^2 + y^2), (x,-3,3), (y,-3,3), start_points=pts)
sage: g += point(pts, color='red')
sage: g
Graphics object consisting of 2 graphics primitives
.. PLOT::
x, y = var('x y')
pts = [[1, 1], [-2, 2], [1, -3/2]]
g = streamline_plot((x + y) / sqrt(x**2 + y**2), (x,-3,3), (y,-3,3), start_points=pts)
g += point(pts, color='red')
sphinx_plot(g)
.. NOTE::
Streamlines currently pass close to ``start_points`` but do
not necessarily pass directly through them. That is part of
the behavior of matplotlib, not an error on your part.
"""
# Parse the function input
if isinstance(f_g, (list, tuple)):
(f,g) = f_g
else:
from sage.functions.all import sqrt
from inspect import isfunction
if isfunction(f_g):
f = lambda x,y: 1 / sqrt(f_g(x, y)**2 + 1)
g = lambda x,y: f_g(x, y) * f(x, y)
else:
f = 1 / sqrt(f_g**2 + 1)
g = f_g * f
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
z, ranges = setup_for_eval_on_grid([f,g], [xrange,yrange], options['plot_points'])
f, g = z
# The density values must be floats
if isinstance(options['density'], (list, tuple)):
options['density'] = [float(x) for x in options['density']]
else:
options['density'] = float(options['density'])
xpos_array, ypos_array, xvec_array, yvec_array = [], [], [], []
for x in xsrange(*ranges[0], include_endpoint=True):
xpos_array.append(x)
for y in xsrange(*ranges[1], include_endpoint=True):
ypos_array.append(y)
xvec_row, yvec_row = [], []
for x in xsrange(*ranges[0], include_endpoint=True):
xvec_row.append(f(x, y))
yvec_row.append(g(x, y))
xvec_array.append(xvec_row)
yvec_array.append(yvec_row)
import numpy
xpos_array = numpy.array(xpos_array, dtype=float)
ypos_array = numpy.array(ypos_array, dtype=float)
xvec_array = numpy.ma.masked_invalid(numpy.array(xvec_array, dtype=float))
yvec_array = numpy.ma.masked_invalid(numpy.array(yvec_array, dtype=float))
if 'start_points' in options:
xstart_array, ystart_array = [], []
for point in options['start_points']:
xstart_array.append(point[0])
ystart_array.append(point[1])
options['start_points'] = numpy.array([xstart_array, ystart_array]).T
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options))
g.add_primitive(StreamlinePlot(xpos_array, ypos_array,
xvec_array, yvec_array, options))
return g