def plot_histogram(self, clr=(0, 0, 1), eps=0.4):
"""
Plots the histogram plot of the sequence, which is assumed to be real
or from a finite field, with a real indexing set I coercible into RR.
Options are clr, which is an RGB value, and eps, which is the spacing between the
bars.
EXAMPLES:
sage: J = range(3)
sage: A = [ZZ(i^2)+1 for i in J]
sage: s = IndexedSequence(A,J)
sage: P = s.plot_histogram()
Now type show(P) to view this in a browser.
"""
#from sage.plot.misc import text
F = self.base_ring() ## elements must be coercible into RR
I = self.index_object()
N = len(I)
S = self.list()
P = [
polygon([[RR(I[i]) - eps, 0], [
RR(I[i]) - eps, RR(S[i])
], [RR(I[i]) + eps, RR(S[i])], [RR(I[i]) + eps, 0], [RR(I[i]), 0]],
rgbcolor=clr) for i in range(N)
]
T = [
text(str(I[i]), (RR(I[i]), -0.8), fontsize=15, rgbcolor=(1, 0, 0))
for i in range(N)
]
return sum(P) + sum(T)
def plot_known(self):
r"""
TESTS::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.function import DFiniteFunction
sage: DiffOps, x, Dx = DifferentialOperators()
sage: f = DFiniteFunction((x^2 + 1)*Dx^2 + 2*x*Dx, [0, 1])
sage: f(-10, 100) # long time
[-1.4711276743037345918528755717...]
sage: f.approx(5, post_transform=Dx) # long time
[0.038461538461538...]
sage: f.plot_known() # long time
Graphics object consisting of ... graphics primitives
"""
g = plot.Graphics()
for center, polys in self._polys.iteritems():
center, rad = self._disk(Point(center, self.dop))
xrange = (center - rad).mid(), (center + rad).mid()
for i, a in enumerate(polys):
# color palette copied from sage.plot.plot.plot
color = plot.Color((0.66666 + i * 0.61803) % 1,
1,
0.4,
space='hsl')
Balls = a.pol.base_ring()
g += plot.plot(lambda x: a.pol(Balls(x)).mid(),
xrange,
color=color)
g += plot.text(str(a.prec), (center, a.pol(center).mid()),
color=color)
for point, ini in self._inivecs.iteritems():
g += plot.point2d((point, 0), size=50)
return g
def plot_histogram(self, clr=(0,0,1), eps = 0.4):
r"""
Plot the histogram plot of the sequence.
The sequence is assumed to be real or from a finite field,
with a real indexing set ``I`` coercible into `\RR`.
Options are ``clr``, which is an RGB value, and ``eps``, which
is the spacing between the bars.
EXAMPLES::
sage: J = range(3)
sage: A = [ZZ(i^2)+1 for i in J]
sage: s = IndexedSequence(A,J)
sage: P = s.plot_histogram()
sage: show(P) # Not tested
"""
# elements must be coercible into RR
I = self.index_object()
N = len(I)
S = self.list()
P = [polygon([[RR(I[i])-eps,0],[RR(I[i])-eps,RR(S[i])],[RR(I[i])+eps,RR(S[i])],[RR(I[i])+eps,0],[RR(I[i]),0]], rgbcolor=clr) for i in range(N)]
T = [text(str(I[i]),(RR(I[i]),-0.8),fontsize=15,rgbcolor=(1,0,0)) for i in range(N)]
return sum(P) + sum(T)
def plot_histogram(self, clr=(0,0,1), eps = 0.4):
r"""
Plot the histogram plot of the sequence.
The sequence is assumed to be real or from a finite field,
with a real indexing set ``I`` coercible into `\RR`.
Options are ``clr``, which is an RGB value, and ``eps``, which
is the spacing between the bars.
EXAMPLES::
sage: J = range(3)
sage: A = [ZZ(i^2)+1 for i in J]
sage: s = IndexedSequence(A,J)
sage: P = s.plot_histogram()
sage: show(P) # Not tested
"""
# elements must be coercible into RR
I = self.index_object()
N = len(I)
S = self.list()
P = [polygon([[RR(I[i])-eps,0],[RR(I[i])-eps,RR(S[i])],[RR(I[i])+eps,RR(S[i])],[RR(I[i])+eps,0],[RR(I[i]),0]], rgbcolor=clr) for i in range(N)]
T = [text(str(I[i]),(RR(I[i]),-0.8),fontsize=15,rgbcolor=(1,0,0)) for i in range(N)]
return sum(P) + sum(T)
def plot_completely_periodic(self):
from sage.plot.all import polygon2d, Graphics, point2d, text
O = self.orbit
G = []
u = self.u # direction (that we put horizontal)
m = matrix(2, [u[1], -u[0], u[1], u[0]])
indices = {}
xmin = xmax = ymin = ymax = 0
for comp in self.decomposition.components():
H = Graphics()
x = O.V2._isomorphic_vector_space.zero()
pts = [x]
below = True
for p in comp.perimeter():
sc = p.saddleConnection()
y = x + m * O.V2._isomorphic_vector_space(
O.V2(p.saddleConnection().vector()))
if p.vertical():
if sc in indices:
i = indices[sc]
else:
i = len(indices) // 2
indices[sc] = i
indices[-sc] = i
if below:
H += text(str(i), (x + y) / 2, color='black')
x = y
xmin = min(xmin, x[0])
xmax = max(xmax, x[0])
ymin = min(ymin, x[1])
ymax = max(ymax, x[1])
pts.append(x)
H += polygon2d(pts, color='blue', alpha=0.3)
H += point2d(pts, color='red', pointsize=20)
G.append(H)
aspect_ratio = float(xmax - xmin) / float(ymax - ymin)
for H in G:
H.set_axes_range(xmin, xmax, ymin, ymax)
H.axes(False)
H.set_aspect_ratio(aspect_ratio)
return G
def plot_labels(self, labels, positions):
r"""
Plot ``labels`` at specified ``positions``.
INPUT:
- ``labels`` -- a string or a list of strings;
- ``positions`` -- a list of points.
OUTPUT:
- a plot.
EXAMPLES::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: print tp.plot_labels("u", [(1.5,0)])
Graphics object consisting of 1 graphics primitive
"""
result = Graphics()
color = self.label_color
extra_options = self.extra_options
zorder = self.label_zorder
font_size = self.font_size
twod = self.dimension <= 2
labels = label_list(labels, len(positions), twod)
for label, position in zip(labels, positions):
if label is None:
continue
if twod:
result += text(label,
position,
color=color,
fontsize=font_size,
zorder=zorder,
**extra_options)
else:
result += text3d(label, position, color=color, **extra_options)
return result
def plot_labels(self, labels, positions):
r"""
Plot ``labels`` at specified ``positions``.
INPUT:
- ``labels`` -- a string or a list of strings;
- ``positions`` -- a list of points.
OUTPUT:
- a plot.
EXAMPLES::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: tp.plot_labels("u", [(1.5,0)])
Graphics object consisting of 1 graphics primitive
"""
result = Graphics()
color = self.label_color
extra_options = self.extra_options
zorder = self.label_zorder
font_size = self.font_size
twod = self.dimension <= 2
labels = label_list(labels, len(positions), twod)
for label, position in zip(labels, positions):
if label is None:
continue
if twod:
result += text(label, position,
color=color, fontsize=font_size,
zorder=zorder, **extra_options)
else:
result += text3d(label, position, color=color, **extra_options)
return result
def plot_histogram(self, clr=(0,0,1),eps = 0.4):
"""
Plots the histogram plot of the sequence, which is assumed to be real
or from a finite field, with a real indexing set I coercible into RR.
Options are clr, which is an RGB value, and eps, which is the spacing between the
bars.
EXAMPLES:
sage: J = range(3)
sage: A = [ZZ(i^2)+1 for i in J]
sage: s = IndexedSequence(A,J)
sage: P = s.plot_histogram()
Now type show(P) to view this in a browser.
"""
#from sage.plot.misc import text
F = self.base_ring() ## elements must be coercible into RR
I = self.index_object()
N = len(I)
S = self.list()
P = [polygon([[RR(I[i])-eps,0],[RR(I[i])-eps,RR(S[i])],[RR(I[i])+eps,RR(S[i])],[RR(I[i])+eps,0],[RR(I[i]),0]], rgbcolor=clr) for i in range(N)]
T = [text(str(I[i]),(RR(I[i]),-0.8),fontsize=15,rgbcolor=(1,0,0)) for i in range(N)]
return sum(P)+sum(T)
def geomrep(M1, B1=None, lineorders1=None, pd=None, sp=False):
"""
Return a sage graphics object containing geometric representation of
matroid M1.
INPUT:
- ``M1`` -- A matroid.
- ``B1`` -- (optional) A list of elements in ``M1.groundset()`` that
correspond to a basis of ``M1`` and will be placed as vertices of the
triangle in the geometric representation of ``M1``.
- ``lineorders1`` -- (optional) A list of ordered lists of elements of
``M1.grondset()`` such that if a line in geometric representation is
setwise same as any of these then points contained will be traversed in
that order thus overriding internal order deciding heuristic.
- ``pd`` - (optional) A dictionary mapping ground set elements to their
(x,y) positions.
- ``sp`` -- (optional) If True, a positioning dictionary and line orders
will be placed in ``M._cached_info``.
OUTPUT:
A sage graphics object of type <class 'sage.plot.graphics.Graphics'> that
corresponds to the geometric representation of the matroid.
EXAMPLES::
sage: from sage.matroids import matroids_plot_helpers
sage: M=matroids.named_matroids.P7()
sage: G=matroids_plot_helpers.geomrep(M)
sage: G.show(xmin=-2, xmax=3, ymin=-2, ymax=3)
sage: M=matroids.named_matroids.P7()
sage: G=matroids_plot_helpers.geomrep(M,lineorders1=[['f','e','d']])
sage: G.show(xmin=-2, xmax=3, ymin=-2, ymax=3)
.. NOTE::
This method does NOT do any checks.
"""
G = Graphics()
# create lists of loops and parallel elements and simplify given matroid
[M, L, P] = slp(M1, pos_dict=pd, B=B1)
if B1 is None:
B1 = list(M.basis())
M._cached_info = M1._cached_info
if M.rank() == 0:
limits = None
loops = L
looptext = ", ".join([str(l) for l in loops])
rectx = -1
recty = -1
rectw = 0.5 + 0.4 * len(loops) + 0.5 # controlled based on len(loops)
recth = 0.6
G += polygon2d(
[[rectx, recty], [rectx, recty + recth],
[rectx + rectw, recty + recth], [rectx + rectw, recty]],
color='black',
fill=False,
thickness=4)
G += text(looptext, (rectx + 0.5, recty + 0.3),
color='black',
fontsize=13)
G += point((rectx + 0.2, recty + 0.3),
color=Color('#BDBDBD'),
size=300,
zorder=2)
G += text('Loop(s)',
(rectx + 0.5 + 0.4 * len(loops) + 0.1, recty + 0.3),
fontsize=13,
color='black')
limits = tracklims(limits, [rectx, rectx + rectw],
[recty, recty + recth])
G.axes(False)
G.axes_range(xmin=limits[0] - 0.5,
xmax=limits[1] + 0.5,
ymin=limits[2] - 0.5,
ymax=limits[3] + 0.5)
return G
elif M.rank() == 1:
if M._cached_info is not None and \
'plot_positions' in M._cached_info.keys() and \
M._cached_info['plot_positions'] is not None:
pts = M._cached_info['plot_positions']
else:
pts = {}
gnd = sorted(M.groundset())
pts[gnd[0]] = (1, float(2) / 3)
G += point((1, float(2) / 3),
size=300,
color=Color('#BDBDBD'),
zorder=2)
pt = [1, float(2) / 3]
if len(P) == 0:
G += text(gnd[0], (float(pt[0]), float(pt[1])),
color='black',
fontsize=13)
pts2 = pts
# track limits [xmin,xmax,ymin,ymax]
pl = [list(x) for x in pts2.values()]
lims = tracklims([None, None, None, None], [pt[0] for pt in pl],
[pt[1] for pt in pl])
elif M.rank() == 2:
nB1 = list(set(list(M.groundset())) - set(B1))
bline = []
for j in nB1:
if M.is_dependent([j, B1[0], B1[1]]):
bline.append(j)
interval = len(bline) + 1
if M._cached_info is not None and \
'plot_positions' in M._cached_info.keys() and \
M._cached_info['plot_positions'] is not None:
pts2 = M._cached_info['plot_positions']
else:
pts2 = {}
pts2[B1[0]] = (0, 0)
pts2[B1[1]] = (2, 0)
lpt = list(pts2[B1[0]])
rpt = list(pts2[B1[1]])
for k in range(len(bline)):
cc = (float(1) / interval) * (k + 1)
pts2[bline[k]] = (cc * lpt[0] + (1 - cc) * rpt[0],
cc * lpt[1] + (1 - cc) * rpt[1])
if sp is True:
M._cached_info['plot_positions'] = pts2
# track limits [xmin,xmax,ymin,ymax]
pl = [list(x) for x in pts2.values()]
lims = tracklims([None, None, None, None], [pt[0] for pt in pl],
[pt[1] for pt in pl])
bline.extend(B1)
ptsx, ptsy, x_i, y_i = createline(pts2, bline, lineorders1)
lims = tracklims(lims, x_i, y_i)
G += line(zip(x_i, y_i), color='black', thickness=3, zorder=1)
pels = [
p for p in pts2.keys() if any([M1.rank([p, q]) == 1 for q in P])
]
allpts = [list(pts2[i]) for i in M.groundset()]
xpts = [float(k[0]) for k in allpts]
ypts = [float(k[1]) for k in allpts]
G += points(zip(xpts, ypts),
color=Color('#BDBDBD'),
size=300,
zorder=2)
for i in pts2:
if i not in pels:
pt = list(pts2[i])
G += text(i, (float(pt[0]), float(pt[1])),
color='black',
fontsize=13)
else:
if M._cached_info is None or \
'plot_positions' not in M._cached_info.keys() or \
M._cached_info['plot_positions'] is None:
(pts, trilines, nontripts,
curvedlines) = it(M1, B1, list(set(M.groundset()) - set(B1)),
list(set(L) | set(P)))
pts2 = addnontripts([B1[0], B1[1], B1[2]], nontripts, pts)
trilines.extend(curvedlines)
else:
pts2 = M._cached_info['plot_positions']
trilines = [
list(set(list(x)).difference(L | P)) for x in M1.flats(2)
if len(list(x)) >= 3
]
pl = [list(x) for x in pts2.values()]
lims = tracklims([None, None, None, None], [pt[0] for pt in pl],
[pt[1] for pt in pl])
j = 0
for ll in trilines:
if len(ll) >= 3:
ptsx, ptsy, x_i, y_i = createline(pts2, ll, lineorders1)
lims = tracklims(lims, x_i, y_i)
G += line(zip(x_i, y_i), color='black', thickness=3, zorder=1)
pels = [
p for p in pts2.keys() if any([M1.rank([p, q]) == 1 for q in P])
]
allpts = [list(pts2[i]) for i in M.groundset()]
xpts = [float(k[0]) for k in allpts]
ypts = [float(k[1]) for k in allpts]
G += points(zip(xpts, ypts),
color=Color('#BDBDBD'),
size=300,
zorder=2)
for i in pts2:
if i not in pels:
pt = list(pts2[i])
G += text(i, (float(pt[0]), float(pt[1])),
color='black',
fontsize=13)
if sp is True:
M1._cached_info['plot_positions'] = pts2
M1._cached_info['plot_lineorders'] = lineorders1
# deal with loops and parallel elements
G, lims = addlp(M1, M, L, P, pts2, G, lims)
G.axes(False)
G.axes_range(xmin=lims[0] - 0.5,
xmax=lims[1] + 0.5,
ymin=lims[2] - 0.5,
ymax=lims[3] + 0.5)
return G
def plot_heap(self):
r"""
Display the Hasse diagram of the heap of ``self``.
The Hasse diagram is rendered in the lattice `S \times \NN`, with
every element `i` in the poset drawn as a point labelled by its label
`s_i`. Every point is placed in the column for its label at a certain
level. The levels start at 0 and the level k of an element `i` is the
maximal number `k` such that the heap contains a chain `i_0\prec
i_1\prec ... \prec i_k` where `i_k=i`. See [Ste1996]_ and [GX2020]_.
OUTPUT: GraphicsObject
EXAMPLES::
sage: FC = CoxeterGroup(['B', 5]).fully_commutative_elements()
sage: FC([3,2,4,3,1]).plot_heap()
Graphics object consisting of 15 graphics primitives
.. PLOT::
:width: 400px
FC = CoxeterGroup(['B', 5]).fully_commutative_elements()
g = FC([3,2,4,3,1]).plot_heap()
sphinx_plot(g)
"""
import sage.plot.all as plot
m = self.parent().coxeter_group().coxeter_matrix()
letters = self.parent().coxeter_group().index_set()
graphics = []
h = self.heap()
levels = h.level_sets()
letters_at_level = [set(self[i] for i in level) for level in levels]
for (level_zero_index, members) in enumerate(levels):
level = level_zero_index + 1
for i in members:
x = self[i]
# Draw the node
graphics.append(
plot.circle((x, level),
0.1,
fill=True,
facecolor='white',
edgecolor='blue',
zorder=1))
graphics.append(
plot.text(str(x), (x, level), color='blue', zorder=2))
neighbors = {z for z in letters if m[x, z] >= 3}
for other in neighbors:
highest_level = max((j + 1 for j in range(level_zero_index)
if other in letters_at_level[j]),
default=None)
if highest_level:
graphics.append(
plot.line([(other, highest_level), (x, level)],
color='black',
zorder=0))
g = sum(graphics)
g.axes(False)
return g
def set_vertices(self, **vertex_options):
"""
Sets the vertex plotting parameters for this GraphPlot. This function
is called by the constructor but can also be called to make updates to
the vertex options of an existing GraphPlot object. Note that the
changes are cumulative.
EXAMPLES::
sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
... (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: GP = g.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
sage: GP.set_vertices(talk=True)
sage: GP.plot()
sage: GP.set_vertices(vertex_colors='pink', vertex_shape='^')
sage: GP.plot()
"""
# Handle base vertex options
voptions = {}
for arg in vertex_options:
self._options[arg] = vertex_options[arg]
# First set defaults for styles
vertex_colors = None
if self._options['talk']:
voptions['markersize'] = 500
if self._options['partition'] is None:
vertex_colors = '#ffffff'
else:
voptions['markersize'] = self._options['vertex_size']
if 'vertex_colors' not in self._options:
if self._options['partition'] is not None:
from sage.plot.colors import rainbow, rgbcolor
partition = self._options['partition']
l = len(partition)
R = rainbow(l)
vertex_colors = {}
for i in range(l):
vertex_colors[R[i]] = partition[i]
elif len(self._graph._boundary) != 0:
vertex_colors = {}
bdy_verts = []
int_verts = []
for v in self._graph.vertex_iterator():
if v in self._graph._boundary:
bdy_verts.append(v)
else:
int_verts.append(v)
vertex_colors['#fec7b8'] = int_verts
vertex_colors['#b3e8ff'] = bdy_verts
elif not vertex_colors:
vertex_colors = '#fec7b8'
else:
vertex_colors = self._options['vertex_colors']
if 'vertex_shape' in self._options:
voptions['marker'] = self._options['vertex_shape']
if self._graph.is_directed():
self._vertex_radius = sqrt(voptions['markersize'] / pi)
self._arrowshorten = 2 * self._vertex_radius
if self._arcdigraph:
self._vertex_radius = sqrt(voptions['markersize'] /
(20500 * pi))
voptions['zorder'] = 7
if not isinstance(vertex_colors, dict):
voptions['facecolor'] = vertex_colors
if self._arcdigraph:
self._plot_components['vertices'] = [
circle(center,
self._vertex_radius,
fill=True,
facecolor=vertex_colors,
clip=False) for center in self._pos.values()
]
else:
self._plot_components['vertices'] = scatter_plot(
self._pos.values(), clip=False, **voptions)
else:
# Color list must be ordered:
pos = []
colors = []
for i in vertex_colors:
pos += [self._pos[j] for j in vertex_colors[i]]
colors += [i] * len(vertex_colors[i])
# If all the vertices have not been assigned a color
if len(self._pos) != len(pos):
from sage.plot.colors import rainbow, rgbcolor
vertex_colors_rgb = [rgbcolor(c) for c in vertex_colors]
for c in rainbow(len(vertex_colors) + 1):
if rgbcolor(c) not in vertex_colors_rgb:
break
leftovers = [j for j in self._pos.values() if j not in pos]
pos += leftovers
colors += [c] * len(leftovers)
if self._arcdigraph:
self._plot_components['vertices'] = [
circle(pos[i],
self._vertex_radius,
fill=True,
facecolor=colors[i],
clip=False) for i in range(len(pos))
]
else:
self._plot_components['vertices'] = scatter_plot(
pos, facecolor=colors, clip=False, **voptions)
if self._options['vertex_labels']:
self._plot_components['vertex_labels'] = []
# TODO: allow text options
for v in self._nodelist:
self._plot_components['vertex_labels'].append(
text(str(v), self._pos[v], rgbcolor=(0, 0, 0), zorder=8))
def geomrep(M1, B1=None, lineorders1=None, pd=None, sp=False):
"""
Return a sage graphics object containing geometric representation of
matroid M1.
INPUT:
- ``M1`` -- A matroid.
- ``B1`` -- (optional) A list of elements in ``M1.groundset()`` that
correspond to a basis of ``M1`` and will be placed as vertices of the
triangle in the geometric representation of ``M1``.
- ``lineorders1`` -- (optional) A list of ordered lists of elements of
``M1.grondset()`` such that if a line in geometric representation is
setwise same as any of these then points contained will be traversed in
that order thus overriding internal order deciding heuristic.
- ``pd`` - (optional) A dictionary mapping ground set elements to their
(x,y) positions.
- ``sp`` -- (optional) If True, a positioning dictionary and line orders
will be placed in ``M._cached_info``.
OUTPUT:
A sage graphics object of type <class 'sage.plot.graphics.Graphics'> that
corresponds to the geometric representation of the matroid.
EXAMPLES::
sage: from sage.matroids import matroids_plot_helpers
sage: M=matroids.named_matroids.P7()
sage: G=matroids_plot_helpers.geomrep(M)
sage: G.show(xmin=-2, xmax=3, ymin=-2, ymax=3)
sage: M=matroids.named_matroids.P7()
sage: G=matroids_plot_helpers.geomrep(M,lineorders1=[['f','e','d']])
sage: G.show(xmin=-2, xmax=3, ymin=-2, ymax=3)
.. NOTE::
This method does NOT do any checks.
"""
G = Graphics()
# create lists of loops and parallel elements and simplify given matroid
[M, L, P] = slp(M1, pos_dict=pd, B=B1)
if B1 is None:
B1 = list(M.basis())
M._cached_info = M1._cached_info
if M.rank() == 0:
limits = None
loops = L
looptext = ", ".join([str(l) for l in loops])
rectx = -1
recty = -1
rectw = 0.5 + 0.4*len(loops) + 0.5 # controlled based on len(loops)
recth = 0.6
G += polygon2d([[rectx, recty], [rectx, recty+recth],
[rectx+rectw, recty+recth], [rectx+rectw, recty]],
color='black', fill=False, thickness=4)
G += text(looptext, (rectx+0.5, recty+0.3), color='black',
fontsize=13)
G += point((rectx+0.2, recty+0.3), color=Color('#BDBDBD'), size=300,
zorder=2)
G += text('Loop(s)', (rectx+0.5+0.4*len(loops)+0.1, recty+0.3),
fontsize=13, color='black')
limits = tracklims(limits, [rectx, rectx+rectw], [recty, recty+recth])
G.axes(False)
G.axes_range(xmin=limits[0]-0.5, xmax=limits[1]+0.5,
ymin=limits[2]-0.5, ymax=limits[3]+0.5)
return G
elif M.rank() == 1:
if M._cached_info is not None and \
'plot_positions' in M._cached_info.keys() and \
M._cached_info['plot_positions'] is not None:
pts = M._cached_info['plot_positions']
else:
pts = {}
gnd = sorted(M.groundset())
pts[gnd[0]] = (1, float(2)/3)
G += point((1, float(2)/3), size=300, color=Color('#BDBDBD'), zorder=2)
pt = [1, float(2)/3]
if len(P) == 0:
G += text(gnd[0], (float(pt[0]), float(pt[1])), color='black',
fontsize=13)
pts2 = pts
# track limits [xmin,xmax,ymin,ymax]
pl = [list(x) for x in pts2.values()]
lims = tracklims([None, None, None, None], [pt[0] for pt in pl],
[pt[1] for pt in pl])
elif M.rank() == 2:
nB1 = list(set(list(M.groundset())) - set(B1))
bline = []
for j in nB1:
if M.is_dependent([j, B1[0], B1[1]]):
bline.append(j)
interval = len(bline)+1
if M._cached_info is not None and \
'plot_positions' in M._cached_info.keys() and \
M._cached_info['plot_positions'] is not None:
pts2 = M._cached_info['plot_positions']
else:
pts2 = {}
pts2[B1[0]] = (0, 0)
pts2[B1[1]] = (2, 0)
lpt = list(pts2[B1[0]])
rpt = list(pts2[B1[1]])
for k in range(len(bline)):
cc = (float(1)/interval)*(k+1)
pts2[bline[k]] = (cc*lpt[0]+(1-cc)*rpt[0],
cc*lpt[1]+(1-cc)*rpt[1])
if sp is True:
M._cached_info['plot_positions'] = pts2
# track limits [xmin,xmax,ymin,ymax]
pl = [list(x) for x in pts2.values()]
lims = tracklims([None, None, None, None], [pt[0] for pt in pl],
[pt[1] for pt in pl])
bline.extend(B1)
ptsx, ptsy, x_i, y_i = createline(pts2, bline, lineorders1)
lims = tracklims(lims, x_i, y_i)
G += line(zip(x_i, y_i), color='black', thickness=3, zorder=1)
pels = [p for p in pts2.keys() if any([M1.rank([p, q]) == 1
for q in P])]
allpts = [list(pts2[i]) for i in M.groundset()]
xpts = [float(k[0]) for k in allpts]
ypts = [float(k[1]) for k in allpts]
G += points(zip(xpts, ypts), color=Color('#BDBDBD'), size=300,
zorder=2)
for i in pts2:
if i not in pels:
pt = list(pts2[i])
G += text(i, (float(pt[0]), float(pt[1])), color='black',
fontsize=13)
else:
if M._cached_info is None or \
'plot_positions' not in M._cached_info.keys() or \
M._cached_info['plot_positions'] is None:
(pts, trilines,
nontripts, curvedlines) = it(M1, B1,
list(set(M.groundset())-set(B1)),
list(set(L) | set(P)))
pts2 = addnontripts([B1[0], B1[1], B1[2]], nontripts, pts)
trilines.extend(curvedlines)
else:
pts2 = M._cached_info['plot_positions']
trilines = [list(set(list(x)).difference(L | P))
for x in M1.flats(2)
if len(list(x)) >= 3]
pl = [list(x) for x in pts2.values()]
lims = tracklims([None, None, None, None], [pt[0] for pt in pl],
[pt[1] for pt in pl])
j = 0
for ll in trilines:
if len(ll) >= 3:
ptsx, ptsy, x_i, y_i = createline(pts2, ll, lineorders1)
lims = tracklims(lims, x_i, y_i)
G += line(zip(x_i, y_i), color='black', thickness=3, zorder=1)
pels = [p for p in pts2.keys() if any([M1.rank([p, q]) == 1
for q in P])]
allpts = [list(pts2[i]) for i in M.groundset()]
xpts = [float(k[0]) for k in allpts]
ypts = [float(k[1]) for k in allpts]
G += points(zip(xpts, ypts), color=Color('#BDBDBD'), size=300,
zorder=2)
for i in pts2:
if i not in pels:
pt = list(pts2[i])
G += text(i, (float(pt[0]), float(pt[1])), color='black',
fontsize=13)
if sp is True:
M1._cached_info['plot_positions'] = pts2
M1._cached_info['plot_lineorders'] = lineorders1
# deal with loops and parallel elements
G, lims = addlp(M1, M, L, P, pts2, G, lims)
G.axes(False)
G.axes_range(xmin=lims[0]-0.5, xmax=lims[1]+0.5, ymin=lims[2]-0.5,
ymax=lims[3]+0.5)
return G
def addlp(M, M1, L, P, ptsdict, G=None, limits=None):
"""
Return a graphics object containing loops (in inset) and parallel elements
of matroid.
INPUT:
- ``M`` -- A matroid.
- ``M1`` -- A simple matroid corresponding to ``M``.
- ``L`` -- List of elements in ``M.groundset()`` that are loops of matroid
``M``.
- ``P`` -- List of elements in ``M.groundset()`` not in
``M.simplify.groundset()`` or ``L``.
- ``ptsdict`` -- A dictionary containing elements in ``M.groundset()`` not
necessarily containing elements of ``L``.
- ``G`` -- (optional) A sage graphics object to which loops and parallel
elements of matroid `M` added .
- ``limits``-- (optional) Current axes limits [xmin,xmax,ymin,ymax].
OUTPUT:
A 2-tuple containing:
1. A sage graphics object containing loops and parallel elements of
matroid ``M``
2. axes limits array
EXAMPLES::
sage: from sage.matroids import matroids_plot_helpers
sage: M=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1],
....: [0, 1, 0, 1, 0, 1, 1,0,0],[0, 0, 1, 1, 1, 0, 1,0,0]])
sage: [M1,L,P]=matroids_plot_helpers.slp(M)
sage: G,lims=matroids_plot_helpers.addlp(M,M1,L,P,{0:(0,0)})
sage: G.show(axes=False)
.. NOTE::
This method does NOT do any checks.
"""
if G is None:
G = Graphics()
# deal with loops
if len(L) > 0:
loops = L
looptext = ", ".join([str(l) for l in loops])
if(limits is None):
rectx = -1
recty = -1
else:
rectx = limits[0]
recty = limits[2]-1
rectw = 0.5 + 0.4*len(loops) + 0.5 # controlled based on len(loops)
recth = 0.6
G += polygon2d([[rectx, recty], [rectx, recty+recth],
[rectx+rectw, recty+recth], [rectx+rectw, recty]],
color='black', fill=False, thickness=4)
G += text(looptext, (rectx+0.5, recty+0.3), color='black',
fontsize=13)
G += point((rectx+0.2, recty+0.3), color=Color('#BDBDBD'), size=300,
zorder=2)
G += text('Loop(s)', (rectx+0.5+0.4*len(loops)+0.1, recty+0.3),
fontsize=13, color='black')
limits = tracklims(limits, [rectx, rectx+rectw], [recty, recty+recth])
# deal with parallel elements
if len(P) > 0:
# create list of lists where inner lists are parallel classes
pcls = []
gnd = sorted(list(M1.groundset()))
for g in gnd:
pcl = [g]
for p in P:
if M.rank([g, p]) == 1:
pcl.extend([p])
pcls.append(pcl)
ext_gnd = list(M.groundset())
for pcl in pcls:
if len(pcl) > 1:
basept = list(ptsdict[pcl[0]])
if len(pcl) <= 2:
# add side by side
ptsdict[pcl[1]] = (basept[0], basept[1]-0.13)
G += points(zip([basept[0]], [basept[1]-0.13]),
color=Color('#BDBDBD'), size=300, zorder=2)
G += text(pcl[0], (float(basept[0]),
float(basept[1])), color='black',
fontsize=13)
G += text(pcl[1], (float(basept[0]),
float(basept[1])-0.13), color='black',
fontsize=13)
limits = tracklims(limits, [basept[0]], [basept[1]-0.13])
else:
# add in a bracket
pce = sorted([str(kk) for kk in pcl])
l = newlabel(set(ext_gnd))
ext_gnd.append(l)
G += text(l+'={ '+", ".join(pce)+' }', (float(basept[0]),
float(basept[1]-0.2)-0.034), color='black',
fontsize=13)
G += text(l, (float(basept[0]),
float(basept[1])), color='black',
fontsize=13)
limits = tracklims(limits, [basept[0]],
[(basept[1]-0.2)-0.034])
return G, limits
def set_edges(self, **edge_options):
"""
Sets the edge (or arrow) plotting parameters for the GraphPlot object. This
function is called by the constructor but can also be called to make updates to
the vertex options of an existing GraphPlot object. Note that the changes are
cumulative.
EXAMPLES::
sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
... (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: GP = g.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
sage: GP.set_edges(edge_style='solid')
sage: GP.plot()
sage: GP.set_edges(edge_color='black')
sage: GP.plot()
sage: d = DiGraph({}, loops=True, multiedges=True, sparse=True)
sage: d.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
... (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: GP = d.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
sage: GP.set_edges(edge_style='solid')
sage: GP.plot()
sage: GP.set_edges(edge_color='black')
sage: GP.plot()
TESTS::
sage: G = Graph("Fooba")
sage: G.show(edge_colors={'red':[(3,6),(2,5)]})
Verify that default edge labels are pretty close to being between the vertices
in some cases where they weren't due to truncating division (trac #10124)::
sage: test_graphs = graphs.FruchtGraph(), graphs.BullGraph()
sage: tol = 0.001
sage: for G in test_graphs:
... E=G.edges()
... for e0, e1, elab in E:
... G.set_edge_label(e0, e1, '%d %d' % (e0, e1))
... gp = G.graphplot(save_pos=True,edge_labels=True)
... vx = gp._plot_components['vertices'][0].xdata
... vy = gp._plot_components['vertices'][0].ydata
... for elab in gp._plot_components['edge_labels']:
... textobj = elab[0]
... x, y, s = textobj.x, textobj.y, textobj.string
... v0, v1 = map(int, s.split())
... vn = vector(((x-(vx[v0]+vx[v1])/2.),y-(vy[v0]+vy[v1])/2.)).norm()
... assert vn < tol
"""
for arg in edge_options:
self._options[arg] = edge_options[arg]
if 'edge_colors' in edge_options: self._options['color_by_label'] = False
# Handle base edge options: thickness, linestyle
eoptions={}
if 'edge_style' in self._options:
eoptions['linestyle'] = self._options['edge_style']
if 'thickness' in self._options:
eoptions['thickness'] = self._options['thickness']
# Set labels param to add labels on the fly
labels = False
if self._options['edge_labels']:
labels = True
self._plot_components['edge_labels'] = []
# Make dict collection of all edges (keep label and edge color)
edges_to_draw = {}
if self._options['color_by_label'] or isinstance(self._options['edge_colors'], dict):
if self._options['color_by_label']: edge_colors = self._graph._color_by_label()
else: edge_colors = self._options['edge_colors']
for color in edge_colors:
for edge in edge_colors[color]:
key = tuple(sorted([edge[0],edge[1]]))
if key == (edge[0],edge[1]): head = 1
else: head = 0
if len(edge) < 3:
label = self._graph.edge_label(edge[0],edge[1])
if isinstance(label, list):
if key in edges_to_draw:
edges_to_draw[key].append((label[-1], color, head))
else:
edges_to_draw[key] = [(label[-1], color, head)]
for i in range(len(label)-1):
edges_to_draw[key].append((label[-1], color, head))
else:
label = edge[2]
if key in edges_to_draw:
edges_to_draw[key].append((label, color, head))
else:
edges_to_draw[key] = [(label, color, head)]
# add unspecified edges in (default color black)
for edge in self._graph.edge_iterator():
key = tuple(sorted([edge[0],edge[1]]))
label = edge[2]
specified = False
if key in edges_to_draw:
for old_label, old_color, old_head in edges_to_draw[key]:
if label == old_label:
specified = True
break
if not specified:
if key == (edge[0],edge[1]): head = 1
else: head = 0
edges_to_draw[key] = [(label, 'black', head)]
else:
for edge in self._graph.edges(sort=True):
key = tuple(sorted([edge[0],edge[1]]))
if key == (edge[0],edge[1]): head = 1
else: head = 0
if key in edges_to_draw:
edges_to_draw[key].append((edge[2], self._options['edge_color'], head))
else:
edges_to_draw[key] = [(edge[2], self._options['edge_color'], head)]
if edges_to_draw:
self._plot_components['edges'] = []
else:
return
# Check for multi-edges or loops
if self._arcs or self._loops:
tmp = edges_to_draw.copy()
dist = self._options['dist']*2.
loop_size = self._options['loop_size']
max_dist = self._options['max_dist']
from sage.functions.all import sqrt
for (a,b) in tmp:
if a == b:
# Loops
distance = dist
local_labels = edges_to_draw.pop((a,b))
if len(local_labels)*dist > max_dist:
distance = float(max_dist)/len(local_labels)
curr_loop_size = loop_size
for i in range(len(local_labels)):
self._plot_components['edges'].append(circle((self._pos[a][0],self._pos[a][1]-curr_loop_size), curr_loop_size, rgbcolor=local_labels[i][1], **eoptions))
if labels:
self._plot_components['edge_labels'].append(text(local_labels[i][0], (self._pos[a][0], self._pos[a][1]-2*curr_loop_size)))
curr_loop_size += distance/4
elif len(edges_to_draw[(a,b)]) > 1:
# Multi-edge
local_labels = edges_to_draw.pop((a,b))
# Compute perpendicular bisector
p1 = self._pos[a]
p2 = self._pos[b]
M = ((p1[0]+p2[0])/2., (p1[1]+p2[1])/2.) # midpoint
if not p1[1] == p2[1]:
S = float(p1[0]-p2[0])/(p2[1]-p1[1]) # perp slope
y = lambda x : S*x-S*M[0]+M[1] # perp bisector line
# f,g are functions of distance d to determine x values
# on line y at d from point M
f = lambda d : sqrt(d**2/(1.+S**2)) + M[0]
g = lambda d : -sqrt(d**2/(1.+S**2)) + M[0]
odd_x = f
even_x = g
if p1[0] == p2[0]:
odd_y = lambda d : M[1]
even_y = odd_y
else:
odd_y = lambda x : y(f(x))
even_y = lambda x : y(g(x))
else:
odd_x = lambda d : M[0]
even_x = odd_x
odd_y = lambda d : M[1] + d
even_y = lambda d : M[1] - d
# We now have the control points for each bezier curve
# in terms of distance parameter d.
# Also note that the label for each edge should be drawn at d/2.
# (This is because we're using the perp bisectors).
distance = dist
if len(local_labels)*dist > max_dist:
distance = float(max_dist)/len(local_labels)
for i in range(len(local_labels)/2):
k = (i+1.0)*distance
if self._arcdigraph:
odd_start = self._polar_hack_for_multidigraph(p1, [odd_x(k),odd_y(k)], self._vertex_radius)[0]
odd_end = self._polar_hack_for_multidigraph([odd_x(k),odd_y(k)], p2, self._vertex_radius)[1]
even_start = self._polar_hack_for_multidigraph(p1, [even_x(k),even_y(k)], self._vertex_radius)[0]
even_end = self._polar_hack_for_multidigraph([even_x(k),even_y(k)], p2, self._vertex_radius)[1]
self._plot_components['edges'].append(arrow(path=[[odd_start,[odd_x(k),odd_y(k)],odd_end]], head=local_labels[2*i][2], zorder=1, rgbcolor=local_labels[2*i][1], **eoptions))
self._plot_components['edges'].append(arrow(path=[[even_start,[even_x(k),even_y(k)],even_end]], head=local_labels[2*i+1][2], zorder=1, rgbcolor=local_labels[2*i+1][1], **eoptions))
else:
self._plot_components['edges'].append(bezier_path([[p1,[odd_x(k),odd_y(k)],p2]],zorder=1, rgbcolor=local_labels[2*i][1], **eoptions))
self._plot_components['edges'].append(bezier_path([[p1,[even_x(k),even_y(k)],p2]],zorder=1, rgbcolor=local_labels[2*i+1][1], **eoptions))
if labels:
j = k/2.0
self._plot_components['edge_labels'].append(text(local_labels[2*i][0],[odd_x(j),odd_y(j)]))
self._plot_components['edge_labels'].append(text(local_labels[2*i+1][0],[even_x(j),even_y(j)]))
if len(local_labels)%2 == 1:
edges_to_draw[(a,b)] = [local_labels[-1]] # draw line for last odd
dir = self._graph.is_directed()
for (a,b) in edges_to_draw:
if self._arcdigraph:
C,D = self._polar_hack_for_multidigraph(self._pos[a], self._pos[b], self._vertex_radius)
self._plot_components['edges'].append(arrow(C,D, rgbcolor=edges_to_draw[(a,b)][0][1], head=edges_to_draw[(a,b)][0][2], **eoptions))
if labels:
self._plot_components['edge_labels'].append(text(str(edges_to_draw[(a,b)][0][0]),[(C[0]+D[0])/2., (C[1]+D[1])/2.]))
elif dir:
self._plot_components['edges'].append(arrow(self._pos[a],self._pos[b], rgbcolor=edges_to_draw[(a,b)][0][1], arrowshorten=self._arrowshorten, head=edges_to_draw[(a,b)][0][2], **eoptions))
else:
self._plot_components['edges'].append(line([self._pos[a],self._pos[b]], rgbcolor=edges_to_draw[(a,b)][0][1], **eoptions))
if labels and not self._arcdigraph:
self._plot_components['edge_labels'].append(text(str(edges_to_draw[(a,b)][0][0]),[(self._pos[a][0]+self._pos[b][0])/2., (self._pos[a][1]+self._pos[b][1])/2.]))
def set_vertices(self, **vertex_options):
"""
Sets the vertex plotting parameters for this GraphPlot. This function
is called by the constructor but can also be called to make updates to
the vertex options of an existing GraphPlot object. Note that the
changes are cumulative.
EXAMPLES::
sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
... (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: GP = g.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
sage: GP.set_vertices(talk=True)
sage: GP.plot()
sage: GP.set_vertices(vertex_colors='pink', vertex_shape='^')
sage: GP.plot()
"""
# Handle base vertex options
voptions = {}
for arg in vertex_options:
self._options[arg] = vertex_options[arg]
# First set defaults for styles
vertex_colors = None
if self._options['talk']:
voptions['markersize'] = 500
if self._options['partition'] is None:
vertex_colors = '#ffffff'
else:
voptions['markersize'] = self._options['vertex_size']
if 'vertex_colors' not in self._options or self._options['vertex_colors'] is None:
if self._options['partition'] is not None:
from sage.plot.colors import rainbow,rgbcolor
partition = self._options['partition']
l = len(partition)
R = rainbow(l)
vertex_colors = {}
for i in range(l):
vertex_colors[R[i]] = partition[i]
elif len(self._graph._boundary) != 0:
vertex_colors = {}
bdy_verts = []
int_verts = []
for v in self._graph.vertex_iterator():
if v in self._graph._boundary:
bdy_verts.append(v)
else:
int_verts.append(v)
vertex_colors['#fec7b8'] = int_verts
vertex_colors['#b3e8ff'] = bdy_verts
elif not vertex_colors:
vertex_colors='#fec7b8'
else:
vertex_colors = self._options['vertex_colors']
if 'vertex_shape' in self._options:
voptions['marker'] = self._options['vertex_shape']
if self._graph.is_directed():
self._vertex_radius = sqrt(voptions['markersize']/pi)
self._arrowshorten = 2*self._vertex_radius
if self._arcdigraph:
self._vertex_radius = sqrt(voptions['markersize']/(20500*pi))
voptions['zorder'] = 7
if not isinstance(vertex_colors, dict):
voptions['facecolor'] = vertex_colors
if self._arcdigraph:
self._plot_components['vertices'] = [circle(center,
self._vertex_radius, fill=True, facecolor=vertex_colors, clip=False)
for center in self._pos.values()]
else:
self._plot_components['vertices'] = scatter_plot(
self._pos.values(), clip=False, **voptions)
else:
# Color list must be ordered:
pos = []
colors = []
for i in vertex_colors:
pos += [self._pos[j] for j in vertex_colors[i]]
colors += [i]*len(vertex_colors[i])
# If all the vertices have not been assigned a color
if len(self._pos)!=len(pos):
from sage.plot.colors import rainbow,rgbcolor
vertex_colors_rgb=[rgbcolor(c) for c in vertex_colors]
for c in rainbow(len(vertex_colors)+1):
if rgbcolor(c) not in vertex_colors_rgb:
break
leftovers=[j for j in self._pos.values() if j not in pos]
pos+=leftovers
colors+=[c]*len(leftovers)
if self._arcdigraph:
self._plot_components['vertices'] = [circle(pos[i],
self._vertex_radius, fill=True, facecolor=colors[i], clip=False)
for i in range(len(pos))]
else:
self._plot_components['vertices'] = scatter_plot(pos,
facecolor=colors, clip=False, **voptions)
if self._options['vertex_labels']:
self._plot_components['vertex_labels'] = []
# TODO: allow text options
for v in self._nodelist:
self._plot_components['vertex_labels'].append(text(str(v),
self._pos[v], rgbcolor=(0,0,0), zorder=8))
def plot_bounds(dop, ini=None, pt=None, eps=None, **kwds):
r"""
EXAMPLES::
sage: from ore_algebra import *
sage: from ore_algebra.analytic.naive_sum import *
sage: Dops, x, Dx = DifferentialOperators()
sage: plot_bounds(Dx - 1, [CBF(1)], CBF(i)/2, RBF(1e-20))
Graphics object consisting of 5 graphics primitives
sage: plot_bounds(x*Dx^3 + 2*Dx^2 + x*Dx, eps=1e-8)
Graphics object consisting of 5 graphics primitives
sage: dop = x*Dx^2 + Dx + x
sage: plot_bounds(dop, eps=1e-8,
....: ini=LogSeriesInitialValues(0, {0: (1, 0)}, dop))
Graphics object consisting of 5 graphics primitives
sage: dop = ((x^2 + 10*x + 50)*Dx^10 + (5/9*x^2 + 50/9*x + 155/9)*Dx^9
....: + (-10/3*x^2 - 100/3*x - 190/3)*Dx^8 + (30*x^2 + 300*x + 815)*Dx^7
....: + (145*x^2 + 1445*x + 3605)*Dx^6 + (5/2*x^2 + 25*x + 115/2)*Dx^5
....: + (20*x^2 + 395/2*x + 1975/4)*Dx^4 + (-5*x^2 - 50*x - 130)*Dx^3
....: + (5/4*x^2 + 25/2*x + 105/4)*Dx^2 + (-20*x^2 - 195*x - 480)*Dx
....: + 5*x - 10)
sage: plot_bounds(dop, pol_part_len=4, bound_inverse="solve", eps=1e-10) # long time
Graphics object consisting of 5 graphics primitives
"""
import sage.plot.all as plot
from sage.all import VectorSpace, QQ, RIF
from ore_algebra.analytic.bounds import abs_min_nonzero_root
if ini is None:
ini = _random_ini(dop)
if pt is None:
rad = abs_min_nonzero_root(dop.leading_coefficient())
pt = QQ(2) if rad == infinity else RIF(rad/2).simplest_rational()
if eps is None:
eps = RBF(1e-50)
logger.info("point: %s", pt)
logger.info("initial values: %s", ini)
recd = []
maj = bounds.DiffOpBound(dop, max_effort=0, **kwds)
series_sum(dop, ini, pt, eps, stride=1, record_bounds_in=recd, maj=maj)
ref_sum = recd[-1][1][0].add_error(recd[-1][2])
recd[-1:] = []
# Note: this won't work well when the errors get close to the double
# precision underflow threshold.
err = [(psum[0]-ref_sum).abs() for n, psum, _ in recd]
large = float(1e200) # plot() is not robust to large values
error_plot_upper = plot.line(
[(n, v.upper()) for (n, v) in enumerate(err)
if abs(float(v.upper())) < large],
color="lightgray", scale="semilogy")
error_plot = plot.line(
[(n, v.lower()) for (n, v) in enumerate(err)
if abs(float(v.lower())) < large],
color="black", scale="semilogy")
bound_plot_lower = plot.line(
[(n, bound.lower()) for n, _, bound in recd
if abs(float(bound.lower())) < large],
color="lightblue", scale="semilogy")
bound_plot = plot.line(
[(n, bound.upper()) for n, _, bound in recd
if abs(float(bound.upper())) < large],
color="blue", scale="semilogy")
title = repr(dop) + " @ x=" + repr(pt)
title = title if len(title) < 80 else title[:77]+"..."
myplot = error_plot_upper + error_plot + bound_plot_lower + bound_plot
ymax = myplot.ymax()
if ymax < float('inf'):
txt = plot.text(title, (myplot.xmax(), ymax),
horizontal_alignment='right', vertical_alignment='top')
myplot += txt
return myplot
def addlp(M, M1, L, P, ptsdict, G=None, limits=None):
"""
Return a graphics object containing loops (in inset) and parallel elements
of matroid.
INPUT:
- ``M`` -- A matroid.
- ``M1`` -- A simple matroid corresponding to ``M``.
- ``L`` -- List of elements in ``M.groundset()`` that are loops of matroid
``M``.
- ``P`` -- List of elements in ``M.groundset()`` not in
``M.simplify.groundset()`` or ``L``.
- ``ptsdict`` -- A dictionary containing elements in ``M.groundset()`` not
necessarily containing elements of ``L``.
- ``G`` -- (optional) A sage graphics object to which loops and parallel
elements of matroid `M` added .
- ``limits``-- (optional) Current axes limits [xmin,xmax,ymin,ymax].
OUTPUT:
A 2-tuple containing:
1. A sage graphics object containing loops and parallel elements of
matroid ``M``
2. axes limits array
EXAMPLES::
sage: from sage.matroids import matroids_plot_helpers
sage: M=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1],
....: [0, 1, 0, 1, 0, 1, 1,0,0],[0, 0, 1, 1, 1, 0, 1,0,0]])
sage: [M1,L,P]=matroids_plot_helpers.slp(M)
sage: G,lims=matroids_plot_helpers.addlp(M,M1,L,P,{0:(0,0)})
sage: G.show(axes=False)
.. NOTE::
This method does NOT do any checks.
"""
if G is None:
G = Graphics()
# deal with loops
if len(L) > 0:
loops = L
looptext = ", ".join([str(l) for l in loops])
if (limits is None):
rectx = -1
recty = -1
else:
rectx = limits[0]
recty = limits[2] - 1
rectw = 0.5 + 0.4 * len(loops) + 0.5 # controlled based on len(loops)
recth = 0.6
G += polygon2d(
[[rectx, recty], [rectx, recty + recth],
[rectx + rectw, recty + recth], [rectx + rectw, recty]],
color='black',
fill=False,
thickness=4)
G += text(looptext, (rectx + 0.5, recty + 0.3),
color='black',
fontsize=13)
G += point((rectx + 0.2, recty + 0.3),
color=Color('#BDBDBD'),
size=300,
zorder=2)
G += text('Loop(s)',
(rectx + 0.5 + 0.4 * len(loops) + 0.1, recty + 0.3),
fontsize=13,
color='black')
limits = tracklims(limits, [rectx, rectx + rectw],
[recty, recty + recth])
# deal with parallel elements
if len(P) > 0:
# create list of lists where inner lists are parallel classes
pcls = []
gnd = sorted(list(M1.groundset()))
for g in gnd:
pcl = [g]
for p in P:
if M.rank([g, p]) == 1:
pcl.extend([p])
pcls.append(pcl)
ext_gnd = list(M.groundset())
for pcl in pcls:
if len(pcl) > 1:
basept = list(ptsdict[pcl[0]])
if len(pcl) <= 2:
# add side by side
ptsdict[pcl[1]] = (basept[0], basept[1] - 0.13)
G += points(zip([basept[0]], [basept[1] - 0.13]),
color=Color('#BDBDBD'),
size=300,
zorder=2)
G += text(pcl[0], (float(basept[0]), float(basept[1])),
color='black',
fontsize=13)
G += text(pcl[1],
(float(basept[0]), float(basept[1]) - 0.13),
color='black',
fontsize=13)
limits = tracklims(limits, [basept[0]], [basept[1] - 0.13])
else:
# add in a bracket
pce = sorted([str(kk) for kk in pcl])
l = newlabel(set(ext_gnd))
ext_gnd.append(l)
G += text(
l + '={ ' + ", ".join(pce) + ' }',
(float(basept[0]), float(basept[1] - 0.2) - 0.034),
color='black',
fontsize=13)
G += text(l, (float(basept[0]), float(basept[1])),
color='black',
fontsize=13)
limits = tracklims(limits, [basept[0]],
[(basept[1] - 0.2) - 0.034])
return G, limits
def set_edges(self, **edge_options):
"""
Sets the edge (or arrow) plotting parameters for the GraphPlot object. This
function is called by the constructor but can also be called to make updates to
the vertex options of an existing GraphPlot object. Note that the changes are
cumulative.
EXAMPLES::
sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
... (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: GP = g.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
sage: GP.set_edges(edge_style='solid')
sage: GP.plot()
sage: GP.set_edges(edge_color='black')
sage: GP.plot()
sage: d = DiGraph({}, loops=True, multiedges=True, sparse=True)
sage: d.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
... (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: GP = d.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
sage: GP.set_edges(edge_style='solid')
sage: GP.plot()
sage: GP.set_edges(edge_color='black')
sage: GP.plot()
TESTS::
sage: G = Graph("Fooba")
sage: G.show(edge_colors={'red':[(3,6),(2,5)]})
Verify that default edge labels are pretty close to being between the vertices
in some cases where they weren't due to truncating division (trac #10124)::
sage: test_graphs = graphs.FruchtGraph(), graphs.BullGraph()
sage: tol = 0.001
sage: for G in test_graphs:
... E=G.edges()
... for e0, e1, elab in E:
... G.set_edge_label(e0, e1, '%d %d' % (e0, e1))
... gp = G.graphplot(save_pos=True,edge_labels=True)
... vx = gp._plot_components['vertices'][0].xdata
... vy = gp._plot_components['vertices'][0].ydata
... for elab in gp._plot_components['edge_labels']:
... textobj = elab[0]
... x, y, s = textobj.x, textobj.y, textobj.string
... v0, v1 = map(int, s.split())
... vn = vector(((x-(vx[v0]+vx[v1])/2.),y-(vy[v0]+vy[v1])/2.)).norm()
... assert vn < tol
"""
for arg in edge_options:
self._options[arg] = edge_options[arg]
if 'edge_colors' in edge_options:
self._options['color_by_label'] = False
# Handle base edge options: thickness, linestyle
eoptions = {}
if 'edge_style' in self._options:
eoptions['linestyle'] = self._options['edge_style']
if 'thickness' in self._options:
eoptions['thickness'] = self._options['thickness']
# Set labels param to add labels on the fly
labels = False
if self._options['edge_labels']:
labels = True
self._plot_components['edge_labels'] = []
# Make dict collection of all edges (keep label and edge color)
edges_to_draw = {}
if self._options['color_by_label'] or isinstance(
self._options['edge_colors'], dict):
if self._options['color_by_label']:
edge_colors = self._graph._color_by_label()
else:
edge_colors = self._options['edge_colors']
for color in edge_colors:
for edge in edge_colors[color]:
key = tuple(sorted([edge[0], edge[1]]))
if key == (edge[0], edge[1]): head = 1
else: head = 0
if len(edge) < 3:
label = self._graph.edge_label(edge[0], edge[1])
if isinstance(label, list):
if key in edges_to_draw:
edges_to_draw[key].append(
(label[-1], color, head))
else:
edges_to_draw[key] = [(label[-1], color, head)]
for i in range(len(label) - 1):
edges_to_draw[key].append(
(label[-1], color, head))
else:
label = edge[2]
if key in edges_to_draw:
edges_to_draw[key].append((label, color, head))
else:
edges_to_draw[key] = [(label, color, head)]
# add unspecified edges in (default color black)
for edge in self._graph.edge_iterator():
key = tuple(sorted([edge[0], edge[1]]))
label = edge[2]
specified = False
if key in edges_to_draw:
for old_label, old_color, old_head in edges_to_draw[key]:
if label == old_label:
specified = True
break
if not specified:
if key == (edge[0], edge[1]): head = 1
else: head = 0
edges_to_draw[key] = [(label, 'black', head)]
else:
for edge in self._graph.edges(sort=True):
key = tuple(sorted([edge[0], edge[1]]))
if key == (edge[0], edge[1]): head = 1
else: head = 0
if key in edges_to_draw:
edges_to_draw[key].append(
(edge[2], self._options['edge_color'], head))
else:
edges_to_draw[key] = [(edge[2],
self._options['edge_color'], head)]
if edges_to_draw:
self._plot_components['edges'] = []
else:
return
# Check for multi-edges or loops
if self._arcs or self._loops:
tmp = edges_to_draw.copy()
dist = self._options['dist'] * 2.
loop_size = self._options['loop_size']
max_dist = self._options['max_dist']
from sage.functions.all import sqrt
for (a, b) in tmp:
if a == b:
# Loops
distance = dist
local_labels = edges_to_draw.pop((a, b))
if len(local_labels) * dist > max_dist:
distance = float(max_dist) / len(local_labels)
curr_loop_size = loop_size
for i in range(len(local_labels)):
self._plot_components['edges'].append(
circle((self._pos[a][0],
self._pos[a][1] - curr_loop_size),
curr_loop_size,
rgbcolor=local_labels[i][1],
**eoptions))
if labels:
self._plot_components['edge_labels'].append(
text(local_labels[i][0],
(self._pos[a][0],
self._pos[a][1] - 2 * curr_loop_size)))
curr_loop_size += distance / 4
elif len(edges_to_draw[(a, b)]) > 1:
# Multi-edge
local_labels = edges_to_draw.pop((a, b))
# Compute perpendicular bisector
p1 = self._pos[a]
p2 = self._pos[b]
M = (
(p1[0] + p2[0]) / 2., (p1[1] + p2[1]) / 2.) # midpoint
if not p1[1] == p2[1]:
S = float(p1[0] - p2[0]) / (p2[1] - p1[1]
) # perp slope
y = lambda x: S * x - S * M[0] + M[
1] # perp bisector line
# f,g are functions of distance d to determine x values
# on line y at d from point M
f = lambda d: sqrt(d**2 / (1. + S**2)) + M[0]
g = lambda d: -sqrt(d**2 / (1. + S**2)) + M[0]
odd_x = f
even_x = g
if p1[0] == p2[0]:
odd_y = lambda d: M[1]
even_y = odd_y
else:
odd_y = lambda x: y(f(x))
even_y = lambda x: y(g(x))
else:
odd_x = lambda d: M[0]
even_x = odd_x
odd_y = lambda d: M[1] + d
even_y = lambda d: M[1] - d
# We now have the control points for each bezier curve
# in terms of distance parameter d.
# Also note that the label for each edge should be drawn at d/2.
# (This is because we're using the perp bisectors).
distance = dist
if len(local_labels) * dist > max_dist:
distance = float(max_dist) / len(local_labels)
for i in range(len(local_labels) / 2):
k = (i + 1.0) * distance
if self._arcdigraph:
odd_start = self._polar_hack_for_multidigraph(
p1, [odd_x(k), odd_y(k)],
self._vertex_radius)[0]
odd_end = self._polar_hack_for_multidigraph(
[odd_x(k), odd_y(k)], p2,
self._vertex_radius)[1]
even_start = self._polar_hack_for_multidigraph(
p1, [even_x(k), even_y(k)],
self._vertex_radius)[0]
even_end = self._polar_hack_for_multidigraph(
[even_x(k), even_y(k)], p2,
self._vertex_radius)[1]
self._plot_components['edges'].append(
arrow(path=[[
odd_start, [odd_x(k), odd_y(k)], odd_end
]],
head=local_labels[2 * i][2],
zorder=1,
rgbcolor=local_labels[2 * i][1],
**eoptions))
self._plot_components['edges'].append(
arrow(path=[[
even_start, [even_x(k),
even_y(k)], even_end
]],
head=local_labels[2 * i + 1][2],
zorder=1,
rgbcolor=local_labels[2 * i + 1][1],
**eoptions))
else:
self._plot_components['edges'].append(
bezier_path(
[[p1, [odd_x(k), odd_y(k)], p2]],
zorder=1,
rgbcolor=local_labels[2 * i][1],
**eoptions))
self._plot_components['edges'].append(
bezier_path(
[[p1, [even_x(k), even_y(k)], p2]],
zorder=1,
rgbcolor=local_labels[2 * i + 1][1],
**eoptions))
if labels:
j = k / 2.0
self._plot_components['edge_labels'].append(
text(local_labels[2 * i][0],
[odd_x(j), odd_y(j)]))
self._plot_components['edge_labels'].append(
text(local_labels[2 * i + 1][0],
[even_x(j), even_y(j)]))
if len(local_labels) % 2 == 1:
edges_to_draw[(a, b)] = [local_labels[-1]
] # draw line for last odd
dir = self._graph.is_directed()
for (a, b) in edges_to_draw:
if self._arcdigraph:
C, D = self._polar_hack_for_multidigraph(
self._pos[a], self._pos[b], self._vertex_radius)
self._plot_components['edges'].append(
arrow(C,
D,
rgbcolor=edges_to_draw[(a, b)][0][1],
head=edges_to_draw[(a, b)][0][2],
**eoptions))
if labels:
self._plot_components['edge_labels'].append(
text(str(edges_to_draw[(a, b)][0][0]),
[(C[0] + D[0]) / 2., (C[1] + D[1]) / 2.]))
elif dir:
self._plot_components['edges'].append(
arrow(self._pos[a],
self._pos[b],
rgbcolor=edges_to_draw[(a, b)][0][1],
arrowshorten=self._arrowshorten,
head=edges_to_draw[(a, b)][0][2],
**eoptions))
else:
self._plot_components['edges'].append(
line([self._pos[a], self._pos[b]],
rgbcolor=edges_to_draw[(a, b)][0][1],
**eoptions))
if labels and not self._arcdigraph:
self._plot_components['edge_labels'].append(
text(str(edges_to_draw[(a, b)][0][0]),
[(self._pos[a][0] + self._pos[b][0]) / 2.,
(self._pos[a][1] + self._pos[b][1]) / 2.]))