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blocks2.py
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blocks2.py
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def convert_table(tab_short, tab_long):
"""
Converts a table with few poles into an equivalent table with many poles.
When tables produced by different methods fail to look the same, it is often
because their polynomials are being multiplied by different positive
prefactors. This adjusts the prefactors so that they are the same.
Parameters
----------
tab_short: A `ConformalBlockTable` where the blocks have a certain number of
poles which is hopefully optimal.
tab_long: A `ConformalBlockTable` with all of the poles that `tab_short` has
plus more.
"""
for l in range(0, len(tab_short.table)):
pole_prod = 1
small_list = tab_short.table[l].poles[:]
for p in tab_long.table[l].poles:
index = get_index_approx(small_list, p)
if index == -1:
pole_prod *= delta - p
tab_short.table[l].poles.append(p)
else:
small_list.remove(small_list[index])
for n in range(0, len(tab_short.table[l].vector)):
tab_short.table[l].vector[n] = tab_short.table[l].vector[n] * pole_prod
tab_short.table[l].vector[n] = tab_short.table[l].vector[n].expand()
def cancel_poles(polynomial_vector):
"""
Checks which roots of a conformal block denominator are also roots of the
numerator. Whenever one is found, a simple factoring is applied.
Parameters
----------
polynomial_vector: The `PolynomialVector` that will be modified in place if
it has superfluous poles.
"""
poles = []
zero_poles = []
for p in polynomial_vector.poles:
if abs(float(p)) > tiny:
poles.append(p)
else:
zero_poles.append(p)
poles = zero_poles + poles
for p in poles:
# We should really make sure the pole is a root of all numerators
# However, this is automatic if it is a root before differentiating
if abs(polynomial_vector.vector[0].subs(delta, p)) < tiny:
polynomial_vector.poles.remove(p)
# A factoring algorithm which works if the zeros are first
for n in range(0, len(polynomial_vector.vector)):
coeffs = coefficients(polynomial_vector.vector[n])
if abs(p) > tiny:
new_coeffs = [coeffs[0] / eval_mpfr(-p, prec)]
for i in range(1, len(coeffs) - 1):
new_coeffs.append((new_coeffs[i - 1] - coeffs[i]) / eval_mpfr(p, prec))
else:
coeffs.remove(coeffs[0])
new_coeffs = coeffs
polynomial_vector.vector[n] = build_polynomial(new_coeffs)
class ConformalBlockTableSeed2:
"""
A class which calculates tables of conformal block derivatives from scratch
using a power series solution of their fourth order differential equation.
Usually, it will not be necessary for the user to call it. Instead,
`ConformalBlockTable` calls it automatically for `m_max = 3`. Note that there
is no `n_max` for this method.
"""
def __init__(self, dim, k_max, l_max, m_max, delta_12 = 0, delta_34 = 0, odd_spins = False):
self.dim = dim
self.k_max = k_max
self.l_max = l_max
self.m_max = m_max
self.delta_12 = delta_12
self.delta_34 = delta_34
self.odd_spins = odd_spins
self.m_order = []
self.n_order = []
self.table = []
if odd_spins:
step = 1
else:
step = 2
pole_set = []
conformal_blocks = []
nu = eval_mpfr((dim / Integer(2)) - 1, prec)
c_2 = (ell * (ell + 2 * nu) + delta * (delta - 2 * nu - 2)) / 2
c_4 = ell * (ell + 2 * nu) * (delta - 1) * (delta - 2 * nu - 1)
delta_prod = delta_12 * delta_34 / (eval_mpfr(-2, prec))
delta_sum = (delta_12 - delta_34) / (eval_mpfr(-2, prec))
if delta_12 == 0 and delta_34 == 0:
effective_power = 2
else:
effective_power = 1
for l in range(0, l_max + 1, step):
poles = []
for k in range(effective_power, k_max + 1, effective_power):
poles.append(eval_mpfr(1 - k - l, prec))
poles.append((2 + 2 * nu - k) / eval_mpfr(2, prec))
poles.append(1 - k + l + 2 * nu)
pole_set.append(poles)
l = 0
while l <= l_max and effective_power == 1:
frob_coeffs = [1]
conformal_blocks.append([])
self.table.append(PolynomialVector([], [l, 0], pole_set[l // step]))
for k in range(1, k_max + 1):
# A good check is to force this code to run for identical scalars too
# This should produce the same blocks as the shorter recursion coming up
recursion_coeffs = [0, 0, 0, 0, 0, 0, 0]
recursion_coeffs[0] += 2 * c_2 * (2 * nu + 1) * (4 * delta_sum + 1) - c_4 + 8 * delta_prod * nu * (2 * nu + 1)
recursion_coeffs[0] -= 2 * (delta + k - 1) * (c_2 * (2 * nu + 1) + 2 * delta_prod * (6 * nu - 1) + 8 * delta_sum * (c_2 + nu - 2 * nu * nu))
recursion_coeffs[0] += 2 * (delta + k - 1) * (delta + k - 2) * (c_2 + nu - 2 * nu * nu + 4 * delta_prod + 12 * delta_sum * (1 - 2 * nu))
recursion_coeffs[0] += 2 * (delta + k - 1) * (delta + k - 2) * (delta + k - 3) * (2 * nu - 1 + 8 * delta_sum)
recursion_coeffs[0] -= 1 * (delta + k - 1) * (delta + k - 2) * (delta + k - 3) * (delta + k - 4)
recursion_coeffs[1] += 3 * c_4 + 2 * c_2 * (4 * delta_sum * (4 * delta_sum + 2 * nu + 1) + 2 * nu - 3) - 8 * delta_prod * (2 * delta_sum * (1 - 6 * nu) + 6 * nu * nu - 5 * nu)
recursion_coeffs[1] -= 2 * (delta + k - 2) * (2 * delta_prod * (16 * delta_sum - 10 * nu + 4) + 8 * delta_sum * (c_2 + nu - 2 * nu * nu) + (1 - 2 * nu) * (c_2 + 2 * nu - 2 + 32 * delta_sum * delta_sum))
recursion_coeffs[1] -= 2 * (delta + k - 2) * (delta + k - 3) * (3 * c_2 + 7 * nu + 2 * nu * nu - 10 + 4 * delta_prod + 4 * delta_sum * (10 * delta_sum + 6 * nu - 3))
recursion_coeffs[1] += 2 * (delta + k - 2) * (delta + k - 3) * (delta + k - 4) * (7 - 2 * nu + 8 * delta_sum)
recursion_coeffs[1] += 3 * (delta + k - 2) * (delta + k - 3) * (delta + k - 4) * (delta + k - 5)
recursion_coeffs[2] += 3 * c_4 + 2 * c_2 * (16 * delta_sum * delta_sum + 2 * nu - 3) + 16 * delta_prod * delta_sum * (8 * delta_sum + 2 * nu + 5)
recursion_coeffs[2] -= 2 * (delta + k - 3) * ((1 - 2 * nu) * (c_2 + 2 * nu - 2 + 32 * delta_sum * delta_sum - 4 * delta_prod) - 8 * delta_sum * (8 * delta_sum * delta_sum + 4 * delta_prod + 4 * nu * nu + 2 * c_2 - 5))
recursion_coeffs[2] -= 2 * (delta + k - 3) * (delta + k - 4) * (8 * delta_prod + 40 * delta_sum * delta_sum + 48 * delta_sum + 3 * c_2 + 2 * nu * nu + 7 * nu - 10)
recursion_coeffs[2] += 2 * (delta + k - 3) * (delta + k - 4) * (delta + k - 5) * (7 - 2 * nu - 16 * delta_sum)
recursion_coeffs[2] += 3 * (delta + k - 3) * (delta + k - 4) * (delta + k - 5) * (delta + k - 6)
recursion_coeffs[3] -= 3 * c_4 + 2 * c_2 * (16 * delta_sum * delta_sum + 2 * nu - 3) + 16 * delta_prod * delta_sum * (8 * delta_sum + 2 * nu + 5)
recursion_coeffs[3] -= 2 * (delta + k - 4) * (12 + 4 * nu - 8 * nu * nu - c_2 * (2 * nu + 5) - 8 * delta_sum * (8 * delta_sum * delta_sum + 8 * delta_sum * nu + 6 * delta_sum + 4 * nu * nu + 2 * c_2 - 5) + 4 * delta_prod * (2 * nu - 5 - 8 * delta_sum))
recursion_coeffs[3] -= 2 * (delta + k - 4) * (delta + k - 5) * (3 * c_2 + 2 * nu * nu + 7 * nu - 10 + 8 * delta_prod + delta_sum * (40 * delta_sum - 18 * nu + 21))
recursion_coeffs[3] -= 2 * (delta + k - 4) * (delta + k - 5) * (delta + k - 6) * (16 * delta_sum + 2 * nu + 11)
recursion_coeffs[3] -= 3 * (delta + k - 4) * (delta + k - 5) * (delta + k - 6) * (delta + k - 7)
recursion_coeffs[4] -= 3 * c_4 + 2 * c_2 * (4 * delta_sum * (4 * delta_sum + 2 * nu + 1) + 2 * nu - 3) - 8 * delta_prod * (2 * delta_sum * (1 - 6 * nu) + 6 * nu * nu - 5 * nu)
recursion_coeffs[4] -= 2 * (delta + k - 5) * (12 + 4 * nu - 8 * nu * nu - c_2 * (2 * nu + 5 - 8 * delta_sum) + 2 * delta_prod * (3 - 10 * nu + 16 * delta_sum) - 8 * delta_sum * (2 * nu * nu + 5 * nu + 3 + 8 * delta_sum * nu + 6 * delta_sum))
recursion_coeffs[4] -= 2 * (delta + k - 5) * (delta + k - 6) * (22 + 5 * nu - 2 * nu * nu - 3 * c_2 - 4 * delta_prod - 4 * delta_sum * (10 * delta_sum + 6 * nu + 9))
recursion_coeffs[4] += 2 * (delta + k - 5) * (delta + k - 6) * (delta + k - 7) * (8 * delta_sum - 2 * nu - 11)
recursion_coeffs[4] -= 3 * (delta + k - 5) * (delta + k - 6) * (delta + k - 7) * (delta + k - 8)
recursion_coeffs[5] -= 2 * c_2 * (2 * nu + 1) * (4 * delta_sum + 1) - c_4 + 8 * delta_prod * nu * (2 * nu + 1)
recursion_coeffs[5] -= 2 * (delta + k - 6) * ((2 * nu + 3) * (c_2 - 2 * nu - 2) + 6 * delta_prod * (2 * nu + 1) + 8 * delta_sum * (c_2 - 2 * nu * nu - 5 * nu - 3))
recursion_coeffs[5] -= 2 * (delta + k - 6) * (delta + k - 7) * (c_2 + 4 * delta_prod - (2 * nu + 3) * (nu + 4 + 12 * delta_sum))
recursion_coeffs[5] += 2 * (delta + k - 6) * (delta + k - 7) * (delta + k - 8) * (2 * nu + 5 + 8 * delta_sum)
recursion_coeffs[5] += 1 * (delta + k - 6) * (delta + k - 7) * (delta + k - 8) * (delta + k - 9)
recursion_coeffs[6] = (k + 2 * nu - 5) * (2 * delta + k - 7) * (delta + k - l - 6) * (delta + k + l + 2 * nu - 6)
recursion_coeffs[5] = recursion_coeffs[5].subs(ell, l)
recursion_coeffs[4] = recursion_coeffs[4].subs(ell, l)
recursion_coeffs[3] = recursion_coeffs[3].subs(ell, l)
recursion_coeffs[2] = recursion_coeffs[2].subs(ell, l)
recursion_coeffs[1] = recursion_coeffs[1].subs(ell, l)
recursion_coeffs[0] = recursion_coeffs[0].subs(ell, l)
pole_prod = 1
frob_coeffs.append(0)
for i in range(0, min(k, 7)):
frob_coeffs[k] += recursion_coeffs[i] * pole_prod * frob_coeffs[k - i - 1] / eval_mpfr(2 * k, prec)
frob_coeffs[k] = frob_coeffs[k].expand()
if i + 1 < min(k, 7):
pole_prod *= (delta - pole_set[l // step][3 * (k - i - 2)]) * (delta - pole_set[l // step][3 * (k - i - 2) + 1]) * (delta - pole_set[l // step][3 * (k - i - 2) + 2])
# We have solved for the Frobenius coefficients times products of poles
# Fix them so that they all carry the same product
pole_prod = 1
for k in range(k_max, -1, -1):
frob_coeffs[k] *= pole_prod
frob_coeffs[k] = frob_coeffs[k].expand()
if k > 0:
pole_prod *= (delta - pole_set[l // step][3 * k - 1]) * (delta - pole_set[l // step][3 * k - 2]) * (delta - pole_set[l // step][3 * k - 3])
conformal_blocks[l // step] = [0] * (m_max + 1)
for k in range(0, k_max + 1):
prod = 1
for m in range(0, m_max + 1):
conformal_blocks[l // step][m] += prod * frob_coeffs[k] * (r_cross ** (k - m))
conformal_blocks[l // step][m] = conformal_blocks[l // step][m].expand()
prod *= (delta + k - m)
l += step
l = 0
while l <= l_max and effective_power == 2:
frob_coeffs = [1]
conformal_blocks.append([])
self.table.append(PolynomialVector([], [l, 0], pole_set[l // step]))
for k in range(2, k_max + 1, 2):
recursion_coeffs = [0, 0, 0]
recursion_coeffs[0] += 3 * c_4 + 2 * c_2 * (2 * nu - 3)
recursion_coeffs[0] += 2 * (delta + k - 2) * (2 * nu - 1) * (c_2 + 2 * nu - 2)
recursion_coeffs[0] += 2 * (delta + k - 2) * (delta + k - 3) * (10 - 7 * nu - 2 * nu * nu - 3 * c_2)
recursion_coeffs[0] += 2 * (delta + k - 2) * (delta + k - 3) * (delta + k - 4) * (7 - 2 * nu)
recursion_coeffs[0] += 3 * (delta + k - 2) * (delta + k - 3) * (delta + k - 4) * (delta + k - 5)
recursion_coeffs[1] += 2 * c_2 * (3 - 2 * nu) - 3 * c_4
recursion_coeffs[1] += 2 * (delta + k - 4) * (c_2 * (2 * nu + 5) + 8 * nu * nu - 4 * nu - 12)
recursion_coeffs[1] += 2 * (delta + k - 4) * (delta + k - 5) * (3 * c_2 + 2 * nu * nu - 5 * nu - 22)
recursion_coeffs[1] -= 2 * (delta + k - 4) * (delta + k - 5) * (delta + k - 6) * (2 * nu + 11)
recursion_coeffs[1] -= 3 * (delta + k - 4) * (delta + k - 5) * (delta + k - 6) * (delta + k - 7)
recursion_coeffs[2] = (k + 2 * nu - 4) * (2 * delta + k - 6) * (delta + k - l - 5) * (delta + k + l + 2 * nu - 5)
recursion_coeffs[1] = recursion_coeffs[1].subs(ell, l)
recursion_coeffs[0] = recursion_coeffs[0].subs(ell, l)
pole_prod = 1
frob_coeffs.append(0)
for i in range(0, min(k / 2, 3)):
frob_coeffs[k / 2] += recursion_coeffs[i] * pole_prod * frob_coeffs[(k / 2) - i - 1] / eval_mpfr(2 * k, prec)
frob_coeffs[k / 2] = frob_coeffs[k / 2].expand()
if i + 1 < min(k / 2, 3):
pole_prod *= (delta - pole_set[l // step][3 * ((k / 2) - i - 2)]) * (delta - pole_set[l // step][3 * ((k / 2) - i - 2) + 1]) * (delta - pole_set[l // step][3 * ((k / 2) - i - 2) + 2])
pole_prod = 1
for k in range(k_max // 2, -1, -1):
frob_coeffs[k] *= pole_prod
frob_coeffs[k] = frob_coeffs[k].expand()
if k > 0:
pole_prod *= (delta - pole_set[l // step][3 * k - 1]) * (delta - pole_set[l // step][3 * k - 2]) * (delta - pole_set[l // step][3 * k - 3])
conformal_blocks[l // step] = [0] * (m_max + 1)
for k in range(0, (k_max // 2) + 1):
prod = 1
for m in range(0, m_max + 1):
conformal_blocks[l // step][m] += prod * frob_coeffs[k] * (r_cross ** (2 * k - m))
conformal_blocks[l // step][m] = conformal_blocks[l // step][m].expand()
prod *= (delta + 2 * k - m)
l += step
(rules1, rules2, self.m_order, self.n_order) = rules(m_max, 0)
chain_rule_single(self.m_order, rules1, self.table, conformal_blocks, lambda l, i: conformal_blocks[l][i])
# Find the superfluous poles (including possible triple poles) to cancel
for l in range(0, len(self.table)):
cancel_poles(self.table[l])