def get_integer(vec):
# Used vec = np.array(vec, dtype=float) before, but does not work for
# [5.00000000e-01, -5.00000000e-01, 2.22044605e-16] in Sigma3[112]
vec = np.round(vec, 12)
vec_sign = np.array([1 if abs(i) == i else -1 for i in vec])
vec = list(abs(vec))
new_vec = []
if 0.0 in vec:
zero_ind = vec.index(0)
vec.pop(zero_ind)
if 0.0 in vec:
new_vec = [get_smallest_multiplier(vec) * i for i in vec]
else:
frac = Fraction(vec[0] / vec[1]).limit_denominator()
new_vec = [frac.numerator, frac.denominator]
new_vec.insert(zero_ind, 0)
else:
for i in range(len(vec) - 1):
frac = Fraction(vec[i] / vec[i + 1]).limit_denominator()
new_vec.extend([frac.numerator, frac.denominator])
if new_vec[1] == new_vec[2]:
new_vec = [new_vec[0], new_vec[1], new_vec[3]]
else:
new_vec = reduce_vector([new_vec[0] * new_vec[2],
new_vec[1] * new_vec[2],
new_vec[3] * new_vec[1]])
assert is_integer(new_vec)
return new_vec * vec_sign
def orthogonalize_csl(csl, axis):
"""
(1) Set the 3rd column of csl same as the rotation axis. The algorithm was
borrowed from gosam project with slight changes. Link to the project:
https://github.com/wojdyr/gosam
(2) Orthogonalize CSL, which is essentially a Gram-Schmidt process. At the
same time, we want to make sure the column vectors of orthogonalized csl
has the smallest value possible. That's why we compared two different ways.
csl = [v1, v2, v3], vi is the column vector
u1 = v3/||v3||, y2 = v1 - (v1 . u1)u1
u2 = y2/||y2||, y3 = v3 - [(v3 . u1)u1 + (v3 . u2)u2]
u3 = y3/||y3||
"""
axis = np.array(axis)
c = solve(csl.transpose(), axis)
if not is_integer(c):
mult = get_smallest_multiplier(c)
c *= mult
c = c.round().astype(int)
ind = min([(i, v) for i, v in enumerate(c) if not np.allclose(v, 0)],
key=lambda x: abs(x[1]))[0]
if ind != 2:
csl[ind], csl[2] = csl[2].copy(), -csl[ind]
c[ind], c[2] = c[2], -c[ind]
csl[2] = np.dot(c, csl)
if c[2] < 0:
csl[1] *= -1
def get_integer(vec):
# Used vec = np.array(vec, dtype=float) before, but does not work for
# [5.00000000e-01, -5.00000000e-01, 2.22044605e-16] in Sigma3[112]
vec = np.round(vec, 12)
vec_sign = np.array([1 if abs(i) == i else -1 for i in vec])
vec = list(abs(vec))
new_vec = []
if 0.0 in vec:
zero_ind = vec.index(0)
vec.pop(zero_ind)
if 0.0 in vec:
new_vec = [get_smallest_multiplier(vec) * i for i in vec]
else:
frac = Fraction(vec[0] / vec[1]).limit_denominator()
new_vec = [frac.numerator, frac.denominator]
new_vec.insert(zero_ind, 0)
else:
for i in range(len(vec) - 1):
frac = Fraction(vec[i] / vec[i + 1]).limit_denominator()
new_vec.extend([frac.numerator, frac.denominator])
if new_vec[1] == new_vec[2]:
new_vec = [new_vec[0], new_vec[1], new_vec[3]]
else:
new_vec = reduce_vector([new_vec[0] * new_vec[2],
new_vec[1] * new_vec[2],
new_vec[3] * new_vec[1]])
assert is_integer(new_vec)
return new_vec * vec_sign
u1 = csl[2] / norm(csl[2])
y2_1 = csl[1] - np.dot(csl[1], u1) * u1
c0_1 = get_integer(y2_1)
y2_2 = csl[0] - np.dot(csl[0], u1) * u1
c0_2 = get_integer(y2_2)
if sum(abs(c0_1)) > sum(abs(c0_2)):
u2 = y2_2 / norm(y2_2)
y3 = csl[1] - np.dot(csl[1], u1) * u1 - np.dot(csl[1], u2) * u2
csl[1] = get_integer(y3)
csl[0] = c0_2
else:
u2 = y2_1 / norm(y2_1)
y3 = csl[0] - np.dot(csl[0], u1) * u1 - np.dot(csl[0], u2) * u2
csl[1] = c0_1
csl[0] = get_integer(y3)
for i in range(3):
for j in range(i + 1, 3):
if not np.allclose(np.dot(csl[i], csl[j]), 0):
raise ValueError("Non-orthogonal basis: %s" % csl)
return csl.round().astype(int)