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 coefficientDerivativeTrace(self, N=0, derivOrder=0, freeIndices=0):
Cd = self.diff(expr=self.Cs[N], order=derivOrder + freeIndices)
result = spatial_diff(expr=Cd,
order=derivOrder) if derivOrder > 0 else Cd
for i in range(derivOrder):
result = np.trace(result, axis1=0, axis2=len(result.shape) - 1)
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):
V = np.tensordot(self.p, self.E, axes=(1, 1)) - np.tensordot(
spatial_diff(self.p, order=1), self.F, axes=((0, 2), (2, 1)))
# First summand. It is transposed into the shape (B,nu)
result = 2 * (self.degP - 1) * np.tensordot(self.Cs[2], V, axes=(0, 1))
# Second summand
result -= np.tensordot(self.Cs[1],
self.diff(self.M, order=0),
axes=(0, 0)).transpose()
# Third summand
tmp = self.sumCoefficientDerivativeContraction(self.M, N=1)
if tmp is not None:
result -= tmp
# Fourth 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):
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 coefficientDerivativeContraction(self,
coeff,
N=0,
derivOrder=0,
Aposition=0,
freeIndices=0):
Cshape = self.Cs[N].shape
Cd = self.diff(expr=self.Cs[N], order=derivOrder + freeIndices)
coeffd = spatial_diff(expr=coeff,
order=derivOrder) if derivOrder == 0 else coeff
# Calculate the index positions for the contraction
Caxes = (len(Cshape), ) + tuple(
range(len(Cd.shape) - derivOrder, len(Cd.shape)))
caxes = (Aposition + derivOrder, ) + tuple(range(derivOrder))
result = np.tensordot(Cd, coeffd, axes=(Caxes, caxes))
return result