def allComponents(self, N):
result = 0
# First term
for K in range(N - 1, self.collapse + 1):
Cd = self.diff(self.Cs[0], order=K)
Md = spatial_diff(self.M, order=K - N + 1)
tmp = np.tensordot(Cd,
Md,
axes=(tuple(range(1, K - N + 2)) + (0, ),
tuple(range(0, K - N + 2))))
tmp = symmetrize(tmp, list(range(len(tmp.shape))))
result = result + 2 * binomial(K, N - 1) * tmp
# Second summand
for K in range(N, self.collapse + 1):
for J in range(N + 1, K + 2):
Cd = self.diff(self.Cs[0], order=K)
Md = spatial_diff(self.M, order=K - J + 1)
tmp = np.tensordot(Cd,
Md,
axes=(tuple(range(1, K - J + 2)) + (0, ),
tuple(range(0, K - J + 2))))
tmp = symmetrize(tmp, list(range(len(tmp.shape))))
tmp = spatial_diff(tmp, order=J - N)
for i in range(J - N + 1):
tmp = tmp.trace(0, len(tmp.shape) - 1)
result = result + (-1)**J * binomial(K, J - 1) * binomial(
J, N) * tmp
return result
def allComponents(self, N):
V = np.tensordot(self.E, self.p, axes=(1, 1)) - np.tensordot(
self.F, spatial_diff(self.p, order=1), axes=((2, 0), (1, 1)))
result = (N + 2) * (self.degP - 1) * np.tensordot(
self.Cs[N + 2], V, axes=(0, 0))
# First M term
tmp = np.tensordot(self.Cs[N + 1],
self.diff(self.M, order=0).swapaxes(1, 2),
axes=(0, 0))
result -= (N + 1) * symmetrize(tmp, list(range(N + 1)))
# The other M terms
tmp = self.sumCoefficientDerivativeContraction(self.M,
N=N + 1,
freeIndices=0)
if tmp is not None:
result -= tmp
# Multiply everything we have so far by (N+1)
result = (N + 1) * result
# The sum with the index swapping
Cd = self.diff(self.Cs[N], order=1)
for K in range(0, N + 1):
result -= Cd.swapaxes(K, len(self.Cs.shape) - 2)
# Last term
result += 2 * spatial_diff(self.diff(self.Cs[N], order=2),
order=1).trace(0, N + 2)
return result
def allComponents(self):
# First term
V = np.tensordot(self.p, self.E, axes=(1, 1)) - np.tensordot(
spatial_diff(expr=self.p, order=1), self.F, axes=((0, 2), (2, 1)))
result = 6 * (self.degP - 1) * np.tensordot(self.Cs[3], V, axes=(0, 1))
# Second term. In the M_: term we swap B2 and mu
result -= 4 * symmetrize(
np.tensordot(self.Cs[2],
self.diff(expr=self.M, order=0).swapaxes(1, 2),
axes=(0, 0)), [0, 1])
# Third to fifth term
tmp = self.sumCoefficientDerivativeContraction(self.M, N=2)
if tmp is not None:
result -= 2 * tmp
# Sixth summand. Need to swap the first and second axes to get into the B1 B2 order
result -= self.diff(expr=self.Cs[1], order=1).swapaxes(0, 1)
# Last summand
tmp = self.sumCoefficientDerivativeTrace(N=1,
freeIndices=1,
combinatorial='K+1',
alternatingSign=True)
if tmp is not None:
result -= tmp
return result
def allComponents(self, N):
# The C_A term
tmpA = self.sumCoefficientDerivativeContraction(
self.M, N=1, freeIndices=N, combinatorial='binomial(K+N,K)')
if tmpA is not None:
tmpA = symmetrize(
tmpA, list(range(len(tmpA.shape) - N, len(tmpA.shape))))
# The C term
tmpB = self.sumCoefficientDerivativeTrace(
N=0,
freeIndices=N + 1,
combinatorial='binomial(K+N-1,K)',
alternatingSign=True)
if tmpB is not None:
tmpB = (-1)**N * tmpB
# Return the result
if tmpA is not None and tmpB is not None:
return tmpA + tmpB
elif tmpA is not None:
return tmpA
elif tmpB is not None:
return tmpB
else:
return np.array([0])
def allComponents(self, N):
Cd = self.diff(expr=self.Cs[N], order=2)
# Contract the coefficient with F
result = np.tensordot(Cd, self.F, axes=(len(Cd.shape)-3,0))
# Symmetrize in alpha, beta, gamma
x = len(result.shape)
return symmetrize(result, [x-4, x-3, x-1])
def allComponents(self, N):
result = - np.tensordot(self.Cs[N], np.eye(3), axes=0)
# Subtract the second term
tmp = np.tensordot(self.Cs[N], self.diff(self.F, order=0).transpose(0,3,2,1), axes=(0,0))
tmp = symmetrize(tmp, list(range(N)))
result -= N * tmp
# E terms
tmp = self.sumCoefficientDerivativeContraction(self.E, N=N, freeIndices=1, combinatorial='K+1')
if tmp is not None:
result += tmp
# F terms
tmp = self.sumCoefficientDerivativeContraction(self.F.swapaxes(1,2))
if tmp is not None:
result -= tmp
return result
def allComponents(self, N):
# E terms
result = 0
tmp = self.coefficientDerivativeContraction(self.E,
N=N,
derivOrder=0,
freeIndices=2)
if tmp is not None:
result = tmp
# F terms
tmp = self.sumCoefficientDerivativeContraction(self.F.swapaxes(1, 2),
N=N,
freeIndices=1,
combinatorial='K+1')
if tmp is not None:
tmp = symmetrize(tmp, [len(tmp.shape) - 3, len(tmp.shape) - 2])
result -= tmp
return result
def allComponents(self):
result = 0
# Third summand
for K in range(2, self.collapse + 1):
for J in range(2, K + 1):
Cd = self.diff(self.Cs[0], order=K)
Md = spatial_diff(self.M, order=K - J)
tmp = np.tensordot(Cd,
Md,
axes=(tuple(range(1, K - J + 1)) + (0, ),
tuple(range(0, K - J + 1))))
tmp = symmetrize(tmp, list(range(len(tmp.shape))))
tmp = spatial_diff(tmp, order=J + 1)
for i in range(J + 1):
tmp = tmp.trace(0, len(tmp.shape) - 1)
result = result + (-1)**J * binomial(K, J) * (J - 1) * tmp
return result
def symmetrizeDerivatives(self, contraction, tensor):
# Import the symmetrization method
from prime.utils import symmetrize
# Unfold
Ks, Phis = contraction
# Prepare the result
from copy import deepcopy
result = deepcopy(tensor)
offset = len(Ks)
for i, Phi in enumerate(Phis):
# If there are more than one derivative indices
if Phi.derivs > 1:
indices = list(range(offset + 1, offset + 1 + Phi.derivs))
result = symmetrize(result, indices)
offset = offset + 1 + Phi.derivs
return result
def symmetrizeBlocks(self, contraction, tensor):
# Import the symmetrization method
from prime.utils import symmetrize
# Unfold
Ks, Phis = contraction
# Prepare the result
from copy import deepcopy
result = deepcopy(tensor)
# First the exchange symmetries in the velocity indices
if len(Ks) > 1:
result = symmetrize(result, list(range(len(Ks))))
# Now the Phi part
if len(Phis) > 1:
for i, Phi in enumerate(Phis):
for j in range(i + 1, len(Phis)):
# If the number of derivative indices is different move to the next one
if Phi.derivs != Phis[j].derivs: continue
# Symmetrize in the i-th and j-th block
shape = tuple(range(len(Ks)))
for k, P in enumerate(Phis):
if k == i:
shape = shape + Phis[
j].getAllIndicesAfterContraction(
offset=Phis[j].new_offset)
elif k == j:
shape = shape + Phi.getAllIndicesAfterContraction(
offset=Phi.new_offset)
else:
shape = shape + P.getAllIndicesAfterContraction(
offset=P.new_offset)
result = (result + result.transpose(shape)) / 2
return result
def allComponents(self):
# First summand. It is transposed into the shape (B,mu,nu)
result = 2 * (self.degP-1) * np.tensordot(np.tensordot(self.p, self.F, axes=(1, 1)), self.Cs[2], axes=(1, 0)) \
.transpose((2,0,1))
# Second summand
tmp = self.sumCoefficientDerivativeContraction(self.M,
N=1,
freeIndices=1,
combinatorial='K+1')
if tmp is not None:
result += tmp
# Third summand
tmp = self.sumCoefficientDerivativeTrace(N=1,
freeIndices=2,
combinatorial='(K+2)*(K+1)/2',
alternatingSign=True)
if tmp is not None:
result -= tmp
return symmetrize(result, [1, 2])
def allComponents(self, N):
result = None
# First term
tmp = self.sumCoefficientDerivativeContraction(
self.E, N=1, freeIndices=N, combinatorial='binomial(K+N,K)')
if tmp is not None:
result = tmp
# Second term
tmp = self.sumCoefficientDerivativeContraction(
self.F.swapaxes(1, 2),
N=1,
freeIndices=N - 1,
combinatorial='binomial(K+N-1,K)')
if tmp is not None:
# Symmetrize in the N free indices
tmp = symmetrize(tmp, list(range(1, N + 1)))
if result is None: result = -tmp
else: result -= tmp
return result
def __init__(self, intertwiners, order=3):
self.intertwiners = intertwiners
self.parametrization = intertwiners.parametrization
self.order = order # TODO: check the order of the intertwiner
# Get the constant part and reshape
constInts = self.intertwiners.constant()
constInts = [i.reshape((int(np.prod(i.shape[0:-1])), i.shape[-1])) for i in constInts]
constIntsIdx = self.intertwiners.indices
# TODO:
# So far, this method is more or less a heuristics that
# works for almost all the parametrizations but it is possible
# that strange examples exist where this is not the case.
# Then the exception will be raised. Try to come up with a
# more general way to solve this.
# Calculate the indices of the inverse intertwiners
constInvsIdx = [[(id[0], -id[1]) for id in idx] for idx in self.intertwiners.indices]
for i in range(len(constInvsIdx)):
constInvsIdx[i].insert(0, constInvsIdx[i].pop(-1))
constInvsShapes = [tuple([id[0] for id in idx]) for idx in constInvsIdx]
# For now, sympy does not support rank defficient matrices, so we cannot use it here.
# There is already a fix merged into master, so it should be usable quite soon...
#constInvs = [ np.reshape(np.array(sympy.Matrix(int).pinv()), shape) for int, shape in zip(constInts, constInvsShapes) ]
constInvs = [np.reshape(np.linalg.pinv(int.astype(np.float64)), shape) for int, shape in zip(constInts, constInvsShapes)]
# Check the parametrization
reshapedInvs = [I.reshape((I.shape[0], int(np.prod(I.shape[1:])))) for I in constInvs]
d = np.zeros((len(self.parametrization.dofs), len(self.parametrization.dofs)))
for A,B in zip(constInts, reshapedInvs):
d = d + np.matmul(B.astype(np.float64),A.astype(np.float64))
valid = np.all((d - np.identity(len(self.parametrization.dofs))) < 1e-10)
if not valid:
raise Exception("The given parametrization is invalid. The inverted intertwiners and the constant ones do not contract to the identity.")
# Import the method for symmetrization
from prime.utils import symmetrize
# Get the orders for the
cacheConsts = [ self.intertwiners.order(o) for o in range(order) ]
cacheInvs = [ constInvs ]
# Iteratively construct the inverse
components = constInvs
for N in range(1, order):
# Take the first one
tmp = sum([np.tensordot(constInvs[i], cacheConsts[N][i],
axes=(
tuple(range(1,len(constInvs[i].shape))),
tuple(range(len(constInvs[i].shape)-1))
)) for i in range(len(constInvs))])
tmp = [-np.tensordot(tmp, constInvs[j], axes=((1,), (0,))) for j in range(len(constInvs))]
# Add the others on top
for k in range(1,N):
tmp2 = sum([np.tensordot(cacheInvs[k][i], cacheConsts[N-k][i],
axes=(
tuple(range(1,len(constInvs[i].shape))),
tuple(range(len(constInvs[i].shape)-1))
)) for i in range(len(constInvs))])
tmp2 = [np.tensordot(tmp2, constInvs[j], axes=((k+1,), (0,))) for j in range(len(constInvs))]
tmp2 = [symmetrize(t, list(range(1, k+1)) + list(range(1, N-k+1))) for t in tmp2]
tmp = [x-y for x,y in zip(tmp, tmp2)]
# Transpose
tmp = [t.transpose((0,) + tuple(range(N+1, len(t.shape))) + tuple(range(1,N+1))) for t in tmp]
# Add to the list
cacheInvs.append(tmp)
# Contract with phis
r = tmp
for i in range(N):
r = [np.dot(t, self.parametrization.dofs) for t in r]
# Add to the components
components = [x + y for x, y in zip(components, r)]
# Set the result
self.components = components
self.indices = constInvsIdx