def main6():
c = continued_fraction([Integer(1)] * (8))
v = [(i, c.p(i) / c.q(i)) for i in range(c.length())]
P = point(v, rgbcolor=(0, 0, 1), pointsize=40)
L = line(v, rgbcolor=(0.5, 0.5, 0.5))
L2 = line([(Integer(0), c.value()), (c.length() - Integer(1), c.value())],
thickness=0.5, rgbcolor=(0.7, 0, 0))
(L + L2 + P).save(filename="continued_fraction.png")
print("Graph is saved as continued_fraction.png")
def draw_transformed_triangle_H(A, xmax=20):
r"""
Draw the modular triangle translated by A=[a,b,c,d]
"""
#print "A=",A,type(A)
pi = RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics, line)
from sage.functions.trig import (cos, sin)
x1 = RR(-0.5)
y1 = RR(sqrt(3) / 2)
x2 = RR(0.5)
y2 = RR(sqrt(3) / 2)
a, b, c, d = A #[0,0]; b=A[0,1]; c=A[1,0]; d=A[1,1]
if a < 0:
a = RR(-a)
b = RR(-b)
c = RR(-c)
d = RR(-d)
else:
a = RR(a)
b = RR(b)
c = RR(c)
d = RR(d)
if c == 0: # then this is easier
if a * d <> 0:
a = a / d
b = b / d
L0 = [[
a * cos(pi_3 * RR(i / 100.0)) + b, a * sin(pi_3 * RR(i / 100.0))
] for i in range(100, 201)]
L1 = [[a * x1 + b, a * y1], [a * x1 + b, xmax]]
L2 = [[a * x2 + b, a * y2], [a * x2 + b, xmax]]
L3 = [[a * x2 + b, xmax], [a * x1 + b, xmax]]
c0 = line(L0)
l1 = line(L1)
l2 = line(L2)
l3 = line(L3)
tri = c0 + l1 + l3 + l2
else:
den = (c * x1 + d)**2 + c**2 * y1**2
x1_t = (a * c * (x1**2 + y1**2) + (a * d + b * c) * x1 + b * d) / den
y1_t = y1 / den
den = (c * x2 + d)**2 + c**2 * y2**2
x2_t = (a * c * (x2**2 + y2**2) + (a * d + b * c) * x2 + b * d) / den
y2_t = y2 / den
inf_t = a / c
#print "A=",A
#print "arg1=",x1_t,y1_t,x2_t,y2_t
c0 = _geodesic_between_two_points(x1_t, y1_t, x2_t, y2_t)
#print "arg1=",x1_t,y1_t,inf_t
c1 = _geodesic_between_two_points(x1_t, y1_t, inf_t, 0.)
#print "arg1=",x2_t,y2_t,inf_t
c2 = _geodesic_between_two_points(x2_t, y2_t, inf_t, 0.0)
tri = c0 + c1 + c2
return tri
def getLfunctionPlot(request, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9):
pythonL = generateLfunctionFromUrl(
arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, to_dict(request.args))
if not pythonL:
return ""
plotrange = 30
if hasattr(pythonL, 'plotpoints'):
F = p2sage(pythonL.plotpoints)
plotrange = min(plotrange, F[-1][0]) # F[-1][0] is the highest t-coordinated that we have a value for L
else:
# obsolete, because lfunc_data comes from DB?
L = pythonL.sageLfunction
if not hasattr(L, "hardy_z_function"):
return None
plotStep = .1
if pythonL._Ltype not in ["riemann", "maass", "ellipticmodularform", "ellipticcurve"]:
plotrange = 12
F = [(i, L.hardy_z_function(i).real()) for i in srange(-1*plotrange, plotrange, plotStep)]
interpolation = spline(F)
F_interp = [(i, interpolation(i)) for i in srange(-1*plotrange, plotrange, 0.05)]
p = line(F_interp)
# p = line(F) # temporary hack while the correct interpolation is being implemented
styleLfunctionPlot(p, 10)
fn = tempfile.mktemp(suffix=".png")
p.save(filename=fn)
data = file(fn).read()
os.remove(fn)
return data
def graphical(ao):
r"""
Return a sage graphics representation of the given acyclic orientation.
"""
from sage.all import Graphics, point, line
assert is_orientation(ao)
n, ascents, descents = ao
# compute heights
height = [0]*n
for _ in range(n):
for (i, j) in ascents:
height[j] = max(height[j], height[i]+1)
for (i, j) in descents:
height[i] = max(height[i], height[j]+1)
# compute the extent of each interval
left = range(n)
right = range(n)
for (i, j) in ascents:
right[i] += .75
left[j] -= .75
for (i, j) in descents:
right[i] += .75
left[j] -= .75
result = Graphics()
# put in the ascents and descents
for (i, j) in ascents:
result += line([(i, height[i]), (j, height[j])],
color='red', thickness=2)
for (i, j) in descents:
result += line([(i, height[i]), (j, height[j])],
color='blue', thickness=2)
# then put in the intervals
for i in range(n):
result += line([(left[i], height[i]), (right[i], height[i])],
color='black', thickness=2)
result += point([(i, height[i])],
color='black', size=100)
# reset some annoying options
result.axes(False)
result.set_aspect_ratio(1)
return result
def _geodesic_between_two_points(x1, y1, x2, y2):
r""" Geodesic path between two points hyperbolic upper half-plane
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points(0.1,0.2,0.0,0.5)
"""
pi = RR.pi()
from sage.plot.plot import line
from sage.functions.trig import arcsin
# logging.debug("z1=%s,%s" % (x1,y1))
# logging.debug("z2=%s,%s" % (x2,y2))
if abs(x1 - x2) < 1e-10:
# The line segment [x=x1, y0<= y <= y1]
return line([[x1, y1], [x2, y2]]) # [0,0,x0,infinity]
c = RR(y1 ** 2 - y2 ** 2 + x1 ** 2 - x2 ** 2) / RR(2 * (x1 - x2))
r = RR(sqrt(y1 ** 2 + (x1 - c) ** 2))
r1 = RR(y1 / r)
r2 = RR(y2 / r)
if abs(r1 - 1) < 1e-12:
r1 = RR(1.0)
elif abs(r2 + 1) < 1e-12:
r2 = -RR(1.0)
if abs(r2 - 1) < 1e-12:
r2 = RR(1.0)
elif abs(r2 + 1) < 1e-12:
r2 = -RR(1.0)
if x1 >= c:
t1 = RR(arcsin(r1))
else:
t1 = RR(pi) - RR(arcsin(r1))
if x2 >= c:
t2 = RR(arcsin(r2))
else:
t2 = RR(pi) - arcsin(r2)
# tmid = (t1 + t2) * RR(0.5)
# a0 = min(t1, t2)
# a1 = max(t1, t2)
# logging.debug("c,r=%s,%s" % (c,r))
# logging.debug("t1,t2=%s,%s"%(t1,t2))
return _circ_arc(t1, t2, c, r)
def _draw_funddom(coset_reps,format="S"):
r""" Draw a fundamental domain for G.
INPUT:
- ``format`` -- (default 'Disp') How to present the f.d.
- ``S`` -- Display directly on the screen
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G._draw_funddom()
"""
pi=RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics,line)
from sage.functions.trig import (cos,sin)
g=Graphics()
x1=RR(-0.5) ; y1=RR(sqrt(3 )/2 )
x2=RR(0.5) ; y2=RR(sqrt(3 )/2 )
xmax=RR(20.0)
l1 = line([[x1,y1],[x1,xmax]])
l2 = line([[x2,y2],[x2,xmax]])
l3 = line([[x2,xmax],[x1,xmax]]) # This is added to make a closed contour
c0=_circ_arc(RR(pi/3.0) ,RR(2.0*pi)/RR(3.0) ,0 ,1 ,100 )
tri=c0+l1+l3+l2
g=g+tri
for A in coset_reps:
[a,b,c,d]=A
if(a==1 and b==0 and c==0 and d==1 ):
continue
if(a<0 ):
a=RR(-a); b=RR(-b); c=RR(-c); d=RR(-d)
else:
a=RR(a); b=RR(b); c=RR(c); d=RR(d)
if(c==0 ): # then this is easier
L0 = [[cos(pi_3*RR(i/100.0))+b,sin(pi_3*RR(i/100.0))] for i in range(100 ,201 )]
L1 = [[x1+b,y1],[x1+b,xmax]]
L2 = [[x2+b,y2],[x2+b,xmax]]
L3 = [[x2+b,xmax],[x1+b,xmax]]
c0=line(L0); l1=line(L1); l2=line(L2); l3=line(L3)
tri=c0+l1+l3+l2
g=g+tri
else:
den=(c*x1+d)**2 +c**2 *y1**2
x1_t=(a*c*(x1**2 +y1**2 )+(a*d+b*c)*x1+b*d)/den
y1_t=y1/den
den=(c*x2+d)**2 +c**2 *y2**2
x2_t=(a*c*(x2**2 +y2**2 )+(a*d+b*c)*x2+b*d)/den
y2_t=y2/den
inf_t=a/c
c0=_geodesic_between_two_points(x1_t,y1_t,x2_t,y2_t)
c1=_geodesic_between_two_points(x1_t,y1_t,inf_t,0. )
c2=_geodesic_between_two_points(x2_t,y2_t,inf_t,0.0)
tri=c0+c1+c2
g=g+tri
return g
def _geodesic_between_two_points(x1, y1, x2, y2):
r""" Geodesic path between two points hyperbolic upper half-plane
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points(0.1,0.2,0.0,0.5)
"""
pi = RR.pi()
from sage.plot.plot import line
from sage.functions.trig import arcsin
#print "z1=",x1,y1
#print "z2=",x2,y2
if (abs(x1 - x2) < 1E-10):
# The line segment [x=x1, y0<= y <= y1]
return line([[x1, y1], [x2, y2]]) #[0,0,x0,infinity]
c = RR(y1**2 - y2**2 + x1**2 - x2**2) / RR(2 * (x1 - x2))
r = RR(sqrt(y1**2 + (x1 - c)**2))
r1 = RR(y1 / r)
r2 = RR(y2 / r)
if (abs(r1 - 1) < 1E-12):
r1 = RR(1.0)
elif (abs(r2 + 1) < 1E-12):
r2 = -RR(1.0)
if (abs(r2 - 1) < 1E-12):
r2 = RR(1.0)
elif (abs(r2 + 1) < 1E-12):
r2 = -RR(1.0)
if (x1 >= c):
t1 = RR(arcsin(r1))
else:
t1 = RR(pi) - RR(arcsin(r1))
if (x2 >= c):
t2 = RR(arcsin(r2))
else:
t2 = RR(pi) - arcsin(r2)
tmid = (t1 + t2) * RR(0.5)
a0 = min(t1, t2)
a1 = max(t1, t2)
#print "c,r=",c,r
#print "t1,t2=",t1,t2
return _circ_arc(t1, t2, c, r)
def plot_norms(norms, block_size, bound):
from sage.all import line
from itertools import cycle
n = len(norms[0])
colours = cycle([
"#4D4D4D", "#5DA5DA", "#FAA43A", "#60BD68", "#F17CB0", "#B2912F",
"#B276B2", "#DECF3F", "#F15854"
])
base = colours.next()
g = line(zip(range(n), norms[0]),
legend_label="GSA",
color=base,
frame=True,
axes=False,
transparent=True,
axes_labels=["$i$", "$2\\,\\log_2 \\|\mathbf{b}^*_i\\|$"])
g += line([(n - block_size, 0),
(n - block_size, 0.99 * norms[0][n - block_size - 1])],
color=base,
linestyle='--')
g += line(zip(range(n), norms[1]),
legend_label="$\\|\\mathbf{e}^*_{i}\\|$",
color=colours.next())
g += line(zip(range(n), norms[2]),
legend_label="LLL",
color=colours.next())
for i, _norms in enumerate(norms[3:]):
if _norms[0] <= bound:
thickness = 1.5
else:
thickness = 1
g += line(zip(range(n), _norms),
legend_label="tour %d" % i,
color=colours.next(),
thickness=thickness)
if _norms[0] <= bound:
break
return g
def draw_transformed_triangle_H(A,xmax=20):
r"""
Draw the modular triangle translated by A=[a,b,c,d]
"""
#print "A=",A,type(A)
pi=RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics,line)
from sage.functions.trig import (cos,sin)
x1=RR(-0.5) ; y1=RR(sqrt(3 )/2 )
x2=RR(0.5) ; y2=RR(sqrt(3 )/2 )
a,b,c,d = A #[0,0]; b=A[0,1]; c=A[1,0]; d=A[1,1]
if a<0:
a=RR(-a); b=RR(-b); c=RR(-c); d=RR(-d)
else:
a=RR(a); b=RR(b); c=RR(c); d=RR(d)
if c==0: # then this is easier
if a*d<>0:
a=a/d; b=b/d;
L0 = [[a*cos(pi_3*RR(i/100.0))+b,a*sin(pi_3*RR(i/100.0))] for i in range(100 ,201 )]
L1 = [[a*x1+b,a*y1],[a*x1+b,xmax]]
L2 = [[a*x2+b,a*y2],[a*x2+b,xmax]]
L3 = [[a*x2+b,xmax],[a*x1+b,xmax]]
c0=line(L0); l1=line(L1); l2=line(L2); l3=line(L3)
tri=c0+l1+l3+l2
else:
den=(c*x1+d)**2 +c**2 *y1**2
x1_t=(a*c*(x1**2 +y1**2 )+(a*d+b*c)*x1+b*d)/den
y1_t=y1/den
den=(c*x2+d)**2 +c**2 *y2**2
x2_t=(a*c*(x2**2 +y2**2 )+(a*d+b*c)*x2+b*d)/den
y2_t=y2/den
inf_t=a/c
#print "A=",A
#print "arg1=",x1_t,y1_t,x2_t,y2_t
c0=_geodesic_between_two_points(x1_t,y1_t,x2_t,y2_t)
#print "arg1=",x1_t,y1_t,inf_t
c1=_geodesic_between_two_points(x1_t,y1_t,inf_t,0. )
#print "arg1=",x2_t,y2_t,inf_t
c2=_geodesic_between_two_points(x2_t,y2_t,inf_t,0.0)
tri=c0+c1+c2
return tri
def draw_funddom(coset_reps, format="S"):
r""" Draw a fundamental domain for G.
INPUT:
- ``format`` -- (default 'Disp') How to present the f.d.
- ``S`` -- Display directly on the screen
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G._draw_funddom()
"""
pi = RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics, line)
from sage.functions.trig import (cos, sin)
g = Graphics()
x1 = RR(-0.5)
y1 = RR(sqrt(3) / 2)
x2 = RR(0.5)
y2 = RR(sqrt(3) / 2)
xmax = RR(20.0)
l1 = line([[x1, y1], [x1, xmax]])
l2 = line([[x2, y2], [x2, xmax]])
l3 = line([[x2, xmax], [x1,
xmax]]) # This is added to make a closed contour
c0 = _circ_arc(RR(pi / 3.0), RR(2.0 * pi) / RR(3.0), 0, 1, 100)
tri = c0 + l1 + l3 + l2
g = g + tri
for A in coset_reps:
if list(A) == [1, 0, 0, 1]:
continue
tri = draw_transformed_triangle_H(A, xmax=xmax)
g = g + tri
return g
def draw_funddom(coset_reps,format="S"):
r""" Draw a fundamental domain for G.
INPUT:
- ``format`` -- (default 'Disp') How to present the f.d.
- ``S`` -- Display directly on the screen
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G._draw_funddom()
"""
pi=RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics,line)
from sage.functions.trig import (cos,sin)
g=Graphics()
x1=RR(-0.5) ; y1=RR(sqrt(3 )/2 )
x2=RR(0.5) ; y2=RR(sqrt(3 )/2 )
xmax=RR(20.0)
l1 = line([[x1,y1],[x1,xmax]])
l2 = line([[x2,y2],[x2,xmax]])
l3 = line([[x2,xmax],[x1,xmax]]) # This is added to make a closed contour
c0=_circ_arc(RR(pi/3.0) ,RR(2.0*pi)/RR(3.0) ,0 ,1 ,100 )
tri=c0+l1+l3+l2
g=g+tri
for A in coset_reps:
if list(A)==[1,0,0,1]:
continue
tri=draw_transformed_triangle_H(A,xmax=xmax)
g=g+tri
return g
def _circ_arc(t0, t1, c, r, num_pts=5000):
r""" Circular arc
INPUTS:
- ''t0'' -- starting parameter
- ''t1'' -- ending parameter
- ''c'' -- center point of the circle
- ''r'' -- radius of circle
- ''num_pts'' -- (default 100) number of points on polygon
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: ca=_circ_arc(0.1,0.2,0.0,1.0,100)
"""
from sage.plot.plot import line
from sage.functions.trig import (cos, sin)
t00 = t0
t11 = t1
## To make sure the line is correct we reduce all arguments to the same branch,
## e.g. [0,2pi]
pi = RR.pi()
while (t00 < 0.0):
t00 = t00 + RR(2.0 * pi)
while (t11 < 0):
t11 = t11 + RR(2.0 * pi)
while (t00 > 2 * pi):
t00 = t00 - RR(2.0 * pi)
while (t11 > 2 * pi):
t11 = t11 - RR(2.0 * pi)
xc = CC(c).real()
yc = CC(c).imag()
L0 = [[
RR(r * cos(t00 + i * (t11 - t00) / num_pts)) + xc,
RR(r * sin(t00 + i * (t11 - t00) / num_pts)) + yc
] for i in range(0, num_pts)]
ca = line(L0)
return ca
def show(self, unit=10, labels=True):
from sage.all import circle, text, line, Graphics
pos = self.basic_grid_embedding()
for v, (a, b) in pos.items():
pos[v] = (unit * a, unit * b)
if not labels:
verts = [circle(p, 1, fill=True) for p in pos.values()]
else:
verts = [
text(repr(v), p, fontsize=20, color='black')
for v, p in pos.items()
]
verts += [circle(p, 1.5, fill=False) for p in pos.values()]
edges = [
line([pos[e.tail], pos[e.head]]) for e in self.edges
if e not in self.dummy
]
G = sum(verts + edges, Graphics())
G.axes(False)
return G
def _circ_arc(t0,t1,c,r,num_pts=5000 ):
r""" Circular arc
INPUTS:
- ''t0'' -- starting parameter
- ''t1'' -- ending parameter
- ''c'' -- center point of the circle
- ''r'' -- radius of circle
- ''num_pts'' -- (default 100) number of points on polygon
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: ca=_circ_arc(0.1,0.2,0.0,1.0,100)
"""
from sage.plot.plot import line
from sage.functions.trig import (cos,sin)
t00=t0; t11=t1
## To make sure the line is correct we reduce all arguments to the same branch,
## e.g. [0,2pi]
pi=RR.pi()
while(t00<0.0):
t00=t00+RR(2.0*pi)
while(t11<0):
t11=t11+RR(2.0*pi)
while(t00>2*pi):
t00=t00-RR(2.0*pi)
while(t11>2*pi):
t11=t11-RR(2.0*pi)
xc=CC(c).real()
yc=CC(c).imag()
L0 = [[RR(r*cos(t00+i*(t11-t00)/num_pts))+xc,RR(r*sin(t00+i*(t11-t00)/num_pts))+yc] for i in range(0 ,num_pts)]
ca=line(L0)
return ca
def _geodesic_between_two_points_d(x1, y1, x2, y2, z0=I):
r""" Geodesic path between two points represented in the unit disc
by the map w = (z-I)/(z+I)
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points_d(0.1,0.2,0.0,0.5)
"""
pi = RR.pi()
from sage.plot.plot import line
from sage.functions.trig import (cos, sin)
# First compute the points
if (y1 < 0 or y2 < 0):
raise ValueError, "Need points in the upper half-plane! Got y1=%s, y2=%s" % (
y1, y2)
if (y1 == infinity):
P1 = CC(1)
else:
P1 = CC((x1 + I * y1 - z0) / (x1 + I * y1 - z0.conjugate()))
if (y2 == infinity):
P2 = CC(1)
else:
P2 = CC((x2 + I * y2 - z0) / (x2 + I * y2 - z0.conjugate()))
# First find the endpoints of the completed geodesic in D
if (x1 == x2):
a = CC((x1 - z0) / (x1 - z0.conjugate()))
b = CC(1)
else:
c = RR(y1**2 - y2**2 + x1**2 - x2**2) / RR(2 * (x1 - x2))
r = RR(sqrt(y1**2 + (x1 - c)**2))
a = c - r
b = c + r
a = CC((a - z0) / (a - z0.conjugate()))
b = CC((b - z0) / (b - z0.conjugate()))
if (abs(a + b) < 1E-10): # On a diagonal
return line([[P1.real(), P1.imag()], [P2.real(), P2.imag()]])
th_a = a.argument()
th_b = b.argument()
# Compute the center of the circle in the disc model
if (min(abs(b - 1), abs(b + 1)) < 1E-10
and min(abs(a - 1), abs(a + 1)) > 1E-10):
c = b + I * (1 - b * cos(th_a)) / sin(th_a)
elif (min(abs(b - 1), abs(b + 1)) > 1E-10
and min(abs(a - 1), abs(a + 1)) < 1E-10):
c = a + I * (1 - a * cos(th_b)) / RR(sin(th_b))
else:
cx = (sin(th_b) - sin(th_a)) / sin(th_b - th_a)
c = cx + I * (1 - cx * cos(th_b)) / RR(sin(th_b))
# First find the endpoints of the completed geodesic
r = abs(c - a)
t1 = CC(P1 - c).argument()
t2 = CC(P2 - c).argument()
#print "t1,t2=",t1,t2
return _circ_arc(t1, t2, c, r)
def draw_fundamental_domain(N,
group='Gamma0',
model="H",
axes=None,
filename=None,
**kwds):
r""" Draw fundamental domain
INPUT:
- ''model'' -- (default ''H'')
- ''H'' -- Upper halfplane
- ''D'' -- Disk model
- ''filename''-- filename to print to
- ''**kwds''-- additional arguments to matplotlib
- ''axes'' -- set geometry of output
=[x0,x1,y0,y1] -- restrict figure to [x0,x1]x[y0,y1]
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G.draw_fundamental_domain()
"""
G = eval(group + '(' + str(N) + ')')
#print G
name = "$" + latex(G) + "$"
## need a "nice" set of coset representatives to draw a connected fundamental domain. Only implemented for Gamma_0(N)
coset_reps = nice_coset_reps(G)
if (model == "D"):
g = draw_funddom_d(coset_reps, format, I)
else:
g = draw_funddom(coset_reps, format)
if (axes <> None):
[x0, x1, y0, y1] = axes
elif (model == "D"):
x0 = -1
x1 = 1
y0 = -1.1
y1 = 1
else:
# find the width of the fundamental domain
w = 0 #self.cusp_width(Cusp(Infinity))
wmin = 0
wmax = 1
max_x = RR(0.55)
rho = CC(exp(2 * pi * I / 3))
for V in coset_reps:
## we also compare the real parts of where rho and infinity are mapped
r1 = (V.acton(rho)).real()
if (V[1, 0] <> 0):
inf1 = RR(V[0, 0] / V[1, 0])
else:
inf1 = 0
if (V[1, 0] == 0 and V[0, 0] == 1):
if (V[0, 1] > wmax):
wmax = V[0, 1]
if (V[0, 1] < wmin):
wmin = V[0, 1]
if (max(r1, inf1) > max_x):
max_x = max(r1, inf1)
#print "wmin,wmax=",wmin,wmax
#x0=-1; x1=1; y0=-0.2; y1=1.5
x0 = RR(-max_x)
x1 = RR(max_x)
y0 = RR(-0.15)
y1 = RR(1.5)
## Draw the axes manually (since can't figure out how to do it automatically)
ax = line([[x0, 0.0], [x1, 0.0]], color='black')
#ax = ax + line([[0.0,y0],[0.0,y1]],color='black')
## ticks
ax = ax + line([[-0.5, -0.01], [-0.5, 0.01]], color='black')
ax = ax + line([[0.5, -0.01], [0.5, 0.01]], color='black')
g = g + ax
if model == "H":
t = text(name, (0, -0.1), fontsize=16, color='black')
else:
t = text(name, (0, -1.1), fontsize=16, color='black')
g = g + t
g.set_aspect_ratio(1)
g.set_axes_range(x0, x1, y0, y1)
g.axes(False)
if (filename <> None):
fig = g.matplotlib()
fig.set_canvas(FigureCanvasAgg(fig))
axes = fig.get_axes()[0]
axes.minorticks_off()
axes.set_yticks([])
fig.savefig(filename, **kwds)
else:
return g
def rhf(b_1_sqr, log_vol, dim):
log_b_sqr = log(b_1_sqr)
log_rhf = (0.5 * log_b_sqr - log_vol / dim) / dim
return exp(log_rhf)
profile_bsw18 = [M.get_r(i, i) for i in range(M.d)]
profile_cn11 = [M.get_r(i, i) for i in range(M.d)]
rhfs_bsw18 = []
rhfs_cn11 = []
rhfs_bsw18.append((0, rhf(profile_bsw18[0], log_vol, M.d)))
rhfs_cn11.append((0, rhf(profile_cn11[0], log_vol, M.d)))
for tour in range(1, max_loops + 1):
profile_bsw18 = BSW18.simulate(
profile_bsw18, BKZ.Param(block_size=block_size, max_loops=1))[0]
profile_cn11 = CN11_simulate(profile_cn11,
BKZ.Param(block_size=block_size,
max_loops=1))[0]
rhfs_bsw18.append((tour, rhf(profile_bsw18[0], log_vol, M.d)))
rhfs_cn11.append((tour, rhf(profile_cn11[0], log_vol, M.d)))
g = line(rhfs_bsw18,
legend_label="BSW18",
linestyle="dashed",
title=f"dim = {M.d}, beta = {block_size}",
axes_labels=["tours", "rhf"])
g += line(rhfs_cn11, legend_label="CN11", linestyle="dashed", color="red")
save(g, f"rhf-{M.d}-{block_size}.png", dpi=150)
def draw_fundamental_domain(N,group='Gamma0',model="H",axes=None,filename=None,**kwds):
r""" Draw fundamental domain
INPUT:
- ''model'' -- (default ''H'')
- ''H'' -- Upper halfplane
- ''D'' -- Disk model
- ''filename''-- filename to print to
- ''**kwds''-- additional arguments to matplotlib
- ''axes'' -- set geometry of output
=[x0,x1,y0,y1] -- restrict figure to [x0,x1]x[y0,y1]
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G.draw_fundamental_domain()
"""
G=eval(group+'('+str(N)+')')
#print G
name ="$"+latex(G)+"$"
## need a "nice" set of coset representatives to draw a connected fundamental domain. Only implemented for Gamma_0(N)
coset_reps = nice_coset_reps(G)
if(model=="D"):
g=draw_funddom_d(coset_reps,format,I)
else:
g=draw_funddom(coset_reps,format)
if(axes<>None):
[x0,x1,y0,y1]=axes
elif(model=="D"):
x0=-1 ; x1=1 ; y0=-1.1 ; y1=1
else:
# find the width of the fundamental domain
w=0 #self.cusp_width(Cusp(Infinity))
wmin=0 ; wmax=1
max_x = RR(0.55)
rho = CC( exp(2*pi*I/3))
for V in coset_reps:
## we also compare the real parts of where rho and infinity are mapped
r1 = (V.acton(rho)).real()
if(V[1,0]<>0):
inf1 = RR(V[0,0] / V[1,0])
else:
inf1 = 0
if(V[1 ,0 ]==0 and V[0 ,0 ]==1 ):
if(V[0 ,1 ]>wmax):
wmax=V[0 ,1 ]
if(V[0 ,1 ]<wmin):
wmin=V[0 ,1 ]
if( max(r1,inf1) > max_x):
max_x = max(r1,inf1)
#print "wmin,wmax=",wmin,wmax
#x0=-1; x1=1; y0=-0.2; y1=1.5
x0=RR(-max_x) ; x1=RR(max_x) ; y0=RR(-0.15) ; y1=RR(1.5)
## Draw the axes manually (since can't figure out how to do it automatically)
ax = line([[x0,0.0],[x1,0.0]],color='black')
#ax = ax + line([[0.0,y0],[0.0,y1]],color='black')
## ticks
ax = ax + line([[-0.5,-0.01],[-0.5,0.01]],color='black')
ax = ax + line([[0.5,-0.01],[0.5,0.01]],color='black')
g = g + ax
if model=="H":
t = text(name, (0, -0.1), fontsize=16, color='black')
else:
t = text(name, (0, -1.1), fontsize=16, color='black')
g = g + t
g.set_aspect_ratio(1)
g.set_axes_range(x0,x1,y0,y1)
g.axes(False)
if(filename<>None):
fig = g.matplotlib()
fig.set_canvas(FigureCanvasAgg(fig))
axes = fig.get_axes()[0]
axes.minorticks_off()
axes.set_yticks([])
fig.savefig(filename,**kwds)
else:
return g
def draw_fundamental_domain(N, group="Gamma0", model="H", axes=None, filename=None, **kwds):
r""" Draw fundamental domain
INPUT:
- ''model'' -- (default ''H'')
= ''H'' -- Upper halfplane
= ''D'' -- Disk model
- ''filename''-- filename to print to
- ''**kwds''-- additional arguments to matplotlib
- ''axes'' -- set geometry of output
=[x0,x1,y0,y1] -- restrict figure to [x0,x1]x[y0,y1]
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G.draw_fundamental_domain()
"""
if group.strip() == "Gamma0":
G = Gamma0(N)
elif group.strip() == "Gamma1":
G = Gamma1(N)
elif group.strip() == "Gamma":
G = Gamma(N)
else:
raise ValueError('group must be one of: "Gamma0", "Gamma1", "Gamma"')
s = "$" + latex(G) + "$"
s = s.replace("mbox", "mathrm")
s = s.replace("Bold", "mathbb")
name = s
# name ="$\mbox{SL}_{2}(\mathbb{Z})$"
# need a "nice" set of coset representatives to draw a connected
# fundamental domain. Only implemented for Gamma_0(N)
coset_reps = nice_coset_reps(G)
# if(group=='Gamma0'):
# else:
# coset_reps = list(G.coset_reps())
from matplotlib.backends.backend_agg import FigureCanvasAgg
if model == "D":
g = _draw_funddom_d(coset_reps, format, I)
else:
g = _draw_funddom(coset_reps, format)
if axes is not None:
[x0, x1, y0, y1] = axes
elif model == "D":
x0 = -1
x1 = 1
y0 = -1.1
y1 = 1
else:
# find the width of the fundamental domain
# w = 0 # self.cusp_width(Cusp(Infinity)) FIXME: w is never used
wmin = 0
wmax = 1
max_x = RR(0.55)
rho = CC(exp(2 * pi * I / 3))
for V in coset_reps:
## we also compare the real parts of where rho and infinity are mapped
r1 = (V.acton(rho)).real()
if V[1, 0] != 0:
inf1 = RR(V[0, 0] / V[1, 0])
else:
inf1 = 0
if V[1, 0] == 0 and V[0, 0] == 1:
if V[0, 1] > wmax:
wmax = V[0, 1]
if V[0, 1] < wmin:
wmin = V[0, 1]
if max(r1, inf1) > max_x:
max_x = max(r1, inf1)
logging.debug("wmin,wmax=%s,%s" % (wmin, wmax))
# x0=-1; x1=1; y0=-0.2; y1=1.5
x0 = RR(-max_x)
x1 = RR(max_x)
y0 = RR(-0.15)
y1 = RR(1.5)
## Draw the axes manually (since can't figure out how to do it automatically)
ax = line([[x0, 0.0], [x1, 0.0]], color="black")
# ax = ax + line([[0.0,y0],[0.0,y1]],color='black')
## ticks
ax = ax + line([[-0.5, -0.01], [-0.5, 0.01]], color="black")
ax = ax + line([[0.5, -0.01], [0.5, 0.01]], color="black")
g = g + ax
if model == "H":
t = text(name, (0, -0.1), fontsize=16, color="black")
t = t + text("$ -\\frac{1}{2} $", (-0.5, -0.1), fontsize=12, color="black")
t = t + text("$ \\frac{1}{2} $", (0.5, -0.1), fontsize=12, color="black")
else:
t = text(name, (0, -1.1), fontsize=16, color="black")
g = g + t
g.set_aspect_ratio(1)
g.set_axes_range(x0, x1, y0, y1)
g.axes(False)
if filename is not None:
fig = g.matplotlib()
fig.set_canvas(FigureCanvasAgg(fig))
axes = fig.get_axes()[0]
axes.minorticks_off()
axes.set_yticks([])
fig.savefig(filename, **kwds)
else:
return g
# n_halfs, block_size, max_loops = 75, 60, 50
# n_halfs, block_size, max_loops = 75, 60, 20000
# generate lattice instance
FPLLL.set_random_seed(1337)
q = 2**30
mat = IntegerMatrix.random(2 * n_halfs, "qary", q=q, k=n_halfs)
A = LLL.reduction(mat)
M = GSO.Mat(A)
M.update_gso()
cn11 = CN11_simulate(M, BKZ.Param(block_size=block_size, max_loops=max_loops))
bsw18 = BSW18.simulate(M, BKZ.Param(block_size=block_size,
max_loops=max_loops))
og = og.simulate(M, BKZ.Param(block_size=block_size, max_loops=max_loops))
g = line([(i, log(cn11[0][i])/2 - log(q)/2) for i in range(len(cn11[0]))]) \
+ line([(i, log(bsw18[0][i])/2 - log(q)/2) for i in range(len(bsw18[0]))], color='red', thickness=2) \
+ line([(i, log(og[0][i])/2 - log(q)/2) for i in range(len(og[0]))], color='green')
save(g, "test.png", dpi=150)
log_vol_cn11 = sum(map(lambda x: log(x) / 2, cn11[0]))
log_vol_bsw18 = sum(map(lambda x: log(x) / 2, bsw18[0]))
log_vol_og = sum(map(lambda x: log(x) / 2, og[0]))
smat = matrix(M.d, M.d)
mat.to_matrix(smat)
vol = abs(smat.determinant())
log_vol = log(vol).n()
print("log vol %.6f" % log_vol)
print(" cn11 %.6f" % log_vol_cn11)
def _draw_funddom(coset_reps, format="S"):
r""" Draw a fundamental domain for G.
INPUT:
- ``format`` -- (default 'Disp') How to present the f.d.
- ``S`` -- Display directly on the screen
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G._draw_funddom()
"""
pi = RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics, line)
from sage.functions.trig import (cos, sin)
g = Graphics()
x1 = RR(-0.5)
y1 = RR(sqrt(3) / 2)
x2 = RR(0.5)
y2 = RR(sqrt(3) / 2)
xmax = RR(20.0)
l1 = line([[x1, y1], [x1, xmax]])
l2 = line([[x2, y2], [x2, xmax]])
l3 = line([[x2, xmax], [x1,
xmax]]) # This is added to make a closed contour
c0 = _circ_arc(RR(pi / 3.0), RR(2.0 * pi) / RR(3.0), 0, 1, 100)
tri = c0 + l1 + l3 + l2
g = g + tri
for A in coset_reps:
[a, b, c, d] = A
if (a == 1 and b == 0 and c == 0 and d == 1):
continue
if (a < 0):
a = RR(-a)
b = RR(-b)
c = RR(-c)
d = RR(-d)
else:
a = RR(a)
b = RR(b)
c = RR(c)
d = RR(d)
if (c == 0): # then this is easier
L0 = [[cos(pi_3 * RR(i / 100.0)) + b,
sin(pi_3 * RR(i / 100.0))] for i in range(100, 201)]
L1 = [[x1 + b, y1], [x1 + b, xmax]]
L2 = [[x2 + b, y2], [x2 + b, xmax]]
L3 = [[x2 + b, xmax], [x1 + b, xmax]]
c0 = line(L0)
l1 = line(L1)
l2 = line(L2)
l3 = line(L3)
tri = c0 + l1 + l3 + l2
g = g + tri
else:
den = (c * x1 + d)**2 + c**2 * y1**2
x1_t = (a * c * (x1**2 + y1**2) +
(a * d + b * c) * x1 + b * d) / den
y1_t = y1 / den
den = (c * x2 + d)**2 + c**2 * y2**2
x2_t = (a * c * (x2**2 + y2**2) +
(a * d + b * c) * x2 + b * d) / den
y2_t = y2 / den
inf_t = a / c
c0 = _geodesic_between_two_points(x1_t, y1_t, x2_t, y2_t)
c1 = _geodesic_between_two_points(x1_t, y1_t, inf_t, 0.)
c2 = _geodesic_between_two_points(x2_t, y2_t, inf_t, 0.0)
tri = c0 + c1 + c2
g = g + tri
return g
def draw_fundamental_domain(N,
group='Gamma0',
model="H",
axes=None,
filename=None,
**kwds):
r""" Draw fundamental domain
INPUT:
- ''model'' -- (default ''H'')
= ''H'' -- Upper halfplane
= ''D'' -- Disk model
- ''filename''-- filename to print to
- ''**kwds''-- additional arguments to matplotlib
- ''axes'' -- set geometry of output
=[x0,x1,y0,y1] -- restrict figure to [x0,x1]x[y0,y1]
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G.draw_fundamental_domain()
"""
if group.strip() == 'Gamma0':
G = Gamma0(N)
elif group.strip() == 'Gamma1':
G = Gamma1(N)
elif group.strip() == 'Gamma':
G = Gamma(N)
else:
raise ValueError('group must be one of: "Gamma0", "Gamma1", "Gamma"')
s = "$" + latex(G) + "$"
s = s.replace("mbox", "mathrm")
s = s.replace("Bold", "mathbb")
name = s
# name ="$\mbox{SL}_{2}(\mathbb{Z})$"
# need a "nice" set of coset representatives to draw a connected
# fundamental domain. Only implemented for Gamma_0(N)
coset_reps = nice_coset_reps(G)
# if(group=='Gamma0'):
# else:
# coset_reps = list(G.coset_reps())
from matplotlib.backends.backend_agg import FigureCanvasAgg
if (model == "D"):
g = _draw_funddom_d(coset_reps, format, I)
else:
g = _draw_funddom(coset_reps, format)
if (axes is not None):
[x0, x1, y0, y1] = axes
elif (model == "D"):
x0 = -1
x1 = 1
y0 = -1.1
y1 = 1
else:
# find the width of the fundamental domain
# w = 0 # self.cusp_width(Cusp(Infinity)) FIXME: w is never used
wmin = 0
wmax = 1
max_x = RR(0.55)
rho = CC(exp(2 * pi * I / 3))
for V in coset_reps:
## we also compare the real parts of where rho and infinity are mapped
r1 = (V.acton(rho)).real()
if (V[1, 0] != 0):
inf1 = RR(V[0, 0] / V[1, 0])
else:
inf1 = 0
if (V[1, 0] == 0 and V[0, 0] == 1):
if (V[0, 1] > wmax):
wmax = V[0, 1]
if (V[0, 1] < wmin):
wmin = V[0, 1]
if (max(r1, inf1) > max_x):
max_x = max(r1, inf1)
logging.debug("wmin,wmax=%s,%s" % (wmin, wmax))
# x0=-1; x1=1; y0=-0.2; y1=1.5
x0 = RR(-max_x)
x1 = RR(max_x)
y0 = RR(-0.15)
y1 = RR(1.5)
## Draw the axes manually (since can't figure out how to do it automatically)
ax = line([[x0, 0.0], [x1, 0.0]], color='black')
# ax = ax + line([[0.0,y0],[0.0,y1]],color='black')
## ticks
ax = ax + line([[-0.5, -0.01], [-0.5, 0.01]], color='black')
ax = ax + line([[0.5, -0.01], [0.5, 0.01]], color='black')
g = g + ax
if model == "H":
t = text(name, (0, -0.1), fontsize=16, color='black')
t = t + text(
"$ -\\frac{1}{2} $", (-0.5, -0.1), fontsize=12, color='black')
t = t + text(
"$ \\frac{1}{2} $", (0.5, -0.1), fontsize=12, color='black')
else:
t = text(name, (0, -1.1), fontsize=16, color='black')
g = g + t
g.set_aspect_ratio(1)
g.set_axes_range(x0, x1, y0, y1)
g.axes(False)
if (filename is not None):
fig = g.matplotlib()
fig.set_canvas(FigureCanvasAgg(fig))
axes = fig.get_axes()[0]
axes.minorticks_off()
axes.set_yticks([])
fig.savefig(filename, **kwds)
else:
return g
def _geodesic_between_two_points_d(x1,y1,x2,y2,z0=I):
r""" Geodesic path between two points represented in the unit disc
by the map w = (z-I)/(z+I)
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points_d(0.1,0.2,0.0,0.5)
"""
pi=RR.pi()
from sage.plot.plot import line
from sage.functions.trig import (cos,sin)
# First compute the points
if(y1<0 or y2<0 ):
raise ValueError,"Need points in the upper half-plane! Got y1=%s, y2=%s" %(y1,y2)
if(y1==infinity):
P1=CC(1 )
else:
P1=CC((x1+I*y1-z0)/(x1+I*y1-z0.conjugate()))
if(y2==infinity):
P2=CC(1 )
else:
P2=CC((x2+I*y2-z0)/(x2+I*y2-z0.conjugate()))
# First find the endpoints of the completed geodesic in D
if(x1==x2):
a=CC((x1-z0)/(x1-z0.conjugate()))
b=CC(1 )
else:
c=RR(y1**2 -y2**2 +x1**2 -x2**2 )/RR(2 *(x1-x2))
r=RR(sqrt(y1**2 +(x1-c)**2 ))
a=c-r
b=c+r
a=CC((a-z0)/(a-z0.conjugate()))
b=CC((b-z0)/(b-z0.conjugate()))
if( abs(a+b) < 1E-10 ): # On a diagonal
return line([[P1.real(),P1.imag()],[P2.real(),P2.imag()]])
th_a=a.argument()
th_b=b.argument()
# Compute the center of the circle in the disc model
if( min(abs(b-1 ),abs(b+1 ))< 1E-10 and min(abs(a-1 ),abs(a+1 ))>1E-10 ):
c=b+I*(1 -b*cos(th_a))/sin(th_a)
elif( min(abs(b-1 ),abs(b+1 ))> 1E-10 and min(abs(a-1 ),abs(a+1 ))<1E-10 ):
c=a+I*(1 -a*cos(th_b))/RR(sin(th_b))
else:
cx=(sin(th_b)-sin(th_a))/sin(th_b-th_a)
c=cx+I*(1 -cx*cos(th_b))/RR(sin(th_b))
# First find the endpoints of the completed geodesic
r=abs(c-a)
t1=CC(P1-c).argument()
t2=CC(P2-c).argument()
#print "t1,t2=",t1,t2
return _circ_arc(t1,t2,c,r)
from sage.all import log, exp
from sage.all import line, save, load, identity_matrix, matrix
from fpylll import IntegerMatrix, GSO, LLL, FPLLL, BKZ
from fpylll.tools.bkz_simulator import simulate as CN11_simulate
import BSW18
# n_halfs, block_size, max_loops = 50, 45, 2000
n_halfs, block_size, max_loops = 90, 170, 60
# n_halfs, block_size, max_loops = 75, 60, 50
# n_halfs, block_size, max_loops = 75, 60, 20000
# generate lattice instance
FPLLL.set_random_seed(1337)
q = 2**30
mat = IntegerMatrix.random(2 * n_halfs, "qary", q=q, k=n_halfs)
A = LLL.reduction(mat)
M = GSO.Mat(A)
M.update_gso()
cn11 = CN11_simulate(M, BKZ.Param(block_size=block_size, max_loops=max_loops))
bsw18 = BSW18.simulate(M, BKZ.Param(block_size=block_size,
max_loops=max_loops))
g = line([(i, log(cn11[0][i])/2 - log(q)/2) for i in range(len(cn11[0]))]) \
+ line([(i, log(bsw18[0][i])/2 - log(q)/2) for i in range(len(bsw18[0]))], color='red')
save(g, "test.png", dpi=150)
