def calc_Vsh(self, n, sqrt_r):
"""Generates m.H(V[n][s]) for a given n, used for generating B[n][s]
This is described on p. 14 of arXiv:1103.0936v2 [cond-mat.str-el] for left
gauge fixing. Here, we are using right gauge fixing.
Array slicing and reshaping is used to manipulate the indices as necessary.
Each V[n] directly depends only on A[n] and r[n].
We return the conjugate m.H(V) because we use it in more places than V.
"""
R = sp.zeros((self.D[n], self.q[n], self.D[n-1]), dtype=self.typ, order='C')
for s in xrange(self.q[n]):
R[:,s,:] = m.mmul(sqrt_r, m.H(self.A[n][s]))
R = R.reshape((self.q[n] * self.D[n], self.D[n-1]))
V = m.H(ns.nullspace(m.H(R)))
#print (q[n]*D[n] - D[n-1], q[n]*D[n])
#print V.shape
#print sp.allclose(mat(V) * mat(V).H, sp.eye(q[n]*D[n] - D[n-1]))
#print sp.allclose(mat(V) * mat(Rh).H, 0)
V = V.reshape((self.q[n] * self.D[n] - self.D[n - 1], self.D[n], self.q[n])) #this works with the above form for R
#prepare for using V[s] and already take the adjoint, since we use it more often
Vsh = sp.empty((self.q[n], self.D[n], self.q[n] * self.D[n] - self.D[n - 1]), dtype=self.typ, order=self.odr)
for s in xrange(self.q[n]):
Vsh[s] = m.H(V[:,:,s])
return Vsh
def computeRigidity(self):
"""returns a k-by-n-by-2 basis of velocity vectors
satisfying the constraints of this"""
m = 2 * len(self.fixed) + len(self.edges) + len(self.angles)
if m == 0: m = 1 # still no constraint, but want a nonempty matrix
n = len(self.vertices)
rigidity = numpy.zeros((m, 2 * n))
row = 0
# 2 constraints per fixed point i,
# to make vx_i=0 and vy_i=0,
# i.e. constrain i to have zero velocity:
for i in self.fixed:
rigidity[row, 2 * i] = 1
rigidity[row + 1, 2 * i + 1] = 1
row += 2
# 1 constraint per edge (i,j),
# to make (p_i-p_j).v_i=(p_i-p_j).v_j,
# i.e. constrain velocities of i and j
# to agree along their common edge:
for (i, j) in self.edges:
u = self.vertices[i]
v = self.vertices[j]
dx = u[0] - v[0]
dy = u[1] - v[1]
rigidity[row, 2 * i] = dx
rigidity[row, 2 * i + 1] = dy
rigidity[row, 2 * j] = -dx
rigidity[row, 2 * j + 1] = -dy
row += 1
# 1 constraint per fixed angle:
for (i, j, k) in self.angles:
u = self.vertices[i]
v = self.vertices[j]
w = self.vertices[k]
dxj = u[0] - v[0]
dyj = u[1] - v[1]
dxk = u[0] - w[0]
dyk = u[1] - w[1]
rigidity[row, 2 * i] = dxj + dxk
rigidity[row, 2 * i + 1] = dyj + dyk
rigidity[row, 2 * j] = -dxk
rigidity[row, 2 * j + 1] = -dyk
rigidity[row, 2 * k] = -dxj
rigidity[row, 2 * k + 1] = -dyj
row += 1
# find basis of velocities satisfying constraints:
null = nullspace.nullspace(rigidity)
# reshape to list of 2D vectors:
velocities = []
for v in null:
velocities.append(v.reshape((n, 2)))
return velocities
def computeRigidity(self):
"""returns a k-by-n-by-2 basis of velocity vectors
satisfying the constraints of this"""
m = 2*len(self.fixed) + len(self.edges) + len(self.angles)
if m==0: m=1 # still no constraint, but want a nonempty matrix
n = len(self.vertices)
rigidity = numpy.zeros((m, 2*n))
row = 0
# 2 constraints per fixed point i,
# to make vx_i=0 and vy_i=0,
# i.e. constrain i to have zero velocity:
for i in self.fixed:
rigidity[row, 2*i] = 1
rigidity[row+1, 2*i+1] = 1
row+=2
# 1 constraint per edge (i,j),
# to make (p_i-p_j).v_i=(p_i-p_j).v_j,
# i.e. constrain velocities of i and j
# to agree along their common edge:
for (i,j) in self.edges:
u = self.vertices[i]
v = self.vertices[j]
dx = u[0]-v[0]
dy = u[1]-v[1]
rigidity[row, 2*i] = dx
rigidity[row, 2*i+1] = dy
rigidity[row, 2*j] = -dx
rigidity[row, 2*j+1] = -dy
row+=1
# 1 constraint per fixed angle:
for (i,j,k) in self.angles:
u = self.vertices[i]
v = self.vertices[j]
w = self.vertices[k]
dxj = u[0]-v[0]
dyj = u[1]-v[1]
dxk = u[0]-w[0]
dyk = u[1]-w[1]
rigidity[row, 2*i] = dxj+dxk
rigidity[row, 2*i+1] = dyj+dyk
rigidity[row, 2*j] = -dxk
rigidity[row, 2*j+1] = -dyk
rigidity[row, 2*k] = -dxj
rigidity[row, 2*k+1] = -dyj
row+=1
# find basis of velocities satisfying constraints:
null = nullspace.nullspace(rigidity)
# reshape to list of 2D vectors:
velocities = []
for v in null:
velocities.append(v.reshape((n,2)))
return velocities
def FindIndependentK(permutation, reference, InteractionPairs):
# kList=[(random.randint(0, Nmax), random.randint(0,Nmax)) for i in range(len(InteractionPairs)+1)]
N=len(InteractionPairs)
Matrix=np.zeros((2*N,3*N))
for i in range(2*N):
interaction=int(i/2)+2*N
sign=i%2
Matrix[i,interaction]=-(-1)**sign
Matrix[i, i]=-1
Matrix[i, permutation.index(i)]=1
# print Matrix
vectors = nullspace(Matrix)
# print len(vectors)
# print vectors
freedoms=vectors.shape[1]
if freedoms!=N+1:
print "Warning! Rank is wrong for {0} with \n{1}".format(permutation, vectors)
return vectors