def F(self, y, j):
N, phi, h, v = y
indices = v.j()
return numpy.array([
P(0),
P(0),
self.V(phi, j) / 3. -
sum(h[p] * h[q] + v[p] * v[q] / 3.
for p, q in nloop(indices, 2) if p != j != q and p + q == j),
-self.dVdphi(phi, j) - 3 *
sum(h[p] * v[q]
for p, q in nloop(indices, 2) if p != j != q and p + q == j),
])
def V(self, phi, j):
indices = phi.j()
ans = P(0)
if j == 2:
ans = ans + self.L
if j >= 2:
ans = ans + 1. / 2 * self.m**2 * sum(phi[p] * phi[q]
for p, q in nloop(indices, 2)
if p + q == j - 2)
ans = ans + 1. / 24 * self.l * sum(
phi[p] * phi[q] * phi[r] * phi[s]
for p, q, r, s in nloop(indices, 4) if p + q == j - 2)
return ans
def V(self, phi, j):
indices = phi.j()
ans = P(0)
if j >= 2:
ans = ans + 1. / 2 * self.m**2 * sum(phi[p] * phi[q]
for p, q in nloop(indices, 2)
if p + q == j - 2)
return ans
def dVdphi(self, phi, j):
ans = P(0)
if j >= 2:
ans = ans + self.m**2 * phi[j - 2]
ans = ans + self.l / 6. * sum(phi[p] * phi[q] * phi[r]
for p, q, r in nloop(indices, 3)
if p + q == j - 2)
return ans
def BellC(x, j):
""" Defines the complete ordinary Bell polynomial recursively. """
indices = [i for i in x.j() if F(i) <= j]
indices = [(p, q) for p, q in nloop(indices, 2) if p != 0 and p + q == j]
if indices:
s = sum(p / j * BellC(x, q) * x[p] for p, q in indices if x[p] != P(0))
if s == 0:
return P(0.)
else:
return s
else:
return P(1)