def make_real_sand_matrix(self, x, y):
r"""Make the superoperator matrix representation of
N[X,Y](rho) = (1/2) ( X rho Y† + Y rho X† )
In the basis {Λ_j}, N[X,Y](rho) = N(x,y)_jk rho_k Λ_j where
N(x,y)_jk = Re[x_m (D_mlj + iF_mlj) (y*)_n (D_knl + iF_knl) ]
Λ_j Λ_k = (D_jkl + iF_jkl) Λ_l
x and y are vectorized representations of the operators X and Y stored
in sparse format.
`sparse.tensordot` might decide to return something dense, so the user
should be aware of that.
"""
result_A = sparse.tensordot(x, self.struct, ([0], [0]))
result_B = sparse.tensordot(np.conj(y), self.struct, ([0], [1]))
# sparse.tensordot fails if both arguments are numpy ndarrays, so we
# force the intermediate arrays to be sparse
if type(result_B) == np.ndarray:
result_B = COO.from_numpy(result_B)
if type(result_A) == np.ndarray:
result_A = COO.from_numpy(result_A)
result = sparse_real(sparse.tensordot(result_A, result_B, ([0], [1])))
# We want our result to be dense, to make things predictable from the
# outside.
if type(result) == sparse.coo.COO:
result = result.todense()
return result.real
def make_real_comm_matrix(self, x, y):
r"""Make the superoperator matrix representation of
M[X,Y](rho) = (1/2) ( [X rho, Y†] + [Y, rho X†] )
In the basis {Λ_j}, M[X,Y](rho) = M(x,y)_jk rho_k Λ_j where
M(x,y)_jk = -2 Im[ (y*)_n F_lnj x_m (D_mkl + iF_mkl) ]
Λ_j Λ_k = (D_jkl + iF_jkl) Λ_l
x and y are vectorized representations of the operators X and Y stored
in sparse format.
`sparse.tensordot` might decide to return something dense, so the user
should be aware of that.
"""
struct_imag = sparse_imag(self.struct)
# sparse.tensordot fails if both arguments are numpy ndarrays, so we
# force the intermediate arrays to be sparse
result_A = sparse.tensordot(np.conj(y), struct_imag, ([0], [1]))
result_B = sparse.tensordot(x, self.struct, ([0], [0]))
if type(result_B) == np.ndarray:
result_B = COO.from_numpy(result_B)
if type(result_A) == np.ndarray:
result_A = COO.from_numpy(result_A)
result = -2 * sparse_imag(sparse.tensordot(result_A, result_B,
([0], [1])))
# We want our result to be dense, to make things predictable from the
# outside.
if type(result) == sparse.coo.COO:
result = result.todense()
return result.real
def vectorize(self, op):
sparse_op = COO.from_numpy(op)
result = sparse.tensordot(self.dual, sparse_op, ([1,2], [1,0]))
if type(result) == np.ndarray:
# I want the result stored in a sparse format even if it isn't
# sparse.
result = COO.from_numpy(result)
return result
def __init__(self, dim):
self.dim = dim
self.basis = COO.from_numpy(np.array(gm.get_basis(dim)))
self.sq_norms = COO.from_numpy(np.einsum('jmn,jnm->j',
self.basis.todense(),
self.basis.todense()))
sq_norms_inv = COO.from_numpy(1 / self.sq_norms.todense())
self.dual = self.basis * sq_norms_inv[:,None,None]
self.struct = sparse.tensordot(sparse.tensordot(self.basis, self.basis,
([2], [1])),
self.dual, ([1, 3], [2, 1]))
if type(self.struct) == np.ndarray:
# Sometimes sparse.tensordot returns numpy arrays. We want to force
# it to be sparse, since sparse.tensordot fails when passed two
# numpy arrays.
self.struct = COO.from_numpy(self.struct)
def check_sparse_hamil_comm(sparse_basis, H, rho):
h_vec = sparse_basis.vectorize(H)
rho_vec = sparse_basis.vectorize(rho)
dense_hamil_comm = -1j * mf.comm(H, rho)
hamil_comm_matrix = sparse_basis.make_hamil_comm_matrix(h_vec)
hamil_comm_vec = COO.from_numpy(hamil_comm_matrix @ rho_vec.todense())
assert_almost_equal(np.abs(dense_hamil_comm -
sparse_basis.matrize(hamil_comm_vec)).max(),
0.0, 7)
def check_sparse_real_sand(sparse_basis, X, rho, Y):
x_vec = sparse_basis.vectorize(X)
y_vec = sparse_basis.vectorize(Y)
rho_vec = sparse_basis.vectorize(rho)
dense_real_sand = (X @ rho @ Y.conj().T + Y @ rho @ X.conj().T) / 2
real_sand_matrix = sparse_basis.make_real_sand_matrix(x_vec, y_vec)
real_sand_vec = COO.from_numpy(real_sand_matrix @ rho_vec.todense())
assert_almost_equal(np.abs(dense_real_sand -
sparse_basis.matrize(real_sand_vec)).max(),
0.0, 7)
def check_sparse_real_comm(sparse_basis, X, rho, Y):
x_vec = sparse_basis.vectorize(X)
y_vec = sparse_basis.vectorize(Y)
rho_vec = sparse_basis.vectorize(rho)
dense_real_comm = (mf.comm(X @ rho, Y.conj().T) +
mf.comm(Y, rho @ X.conj().T)) / 2
real_comm_matrix = sparse_basis.make_real_comm_matrix(x_vec, y_vec)
real_comm_vec = COO.from_numpy(real_comm_matrix @ rho_vec.todense())
assert_almost_equal(np.abs(dense_real_comm -
sparse_basis.matrize(real_comm_vec)).max(),
0.0, 7)
def test_original_is_copied(self, shift, ax):
xs = sparse.random((10, 10), density=0.5)
xc = COO(np.copy(xs.coords), np.copy(xs.data), shape=xs.shape)
sparse.roll(xs, shift, axis=ax)
assert_eq(xs, xc)
def test_coord_dtype():
s = sparse.random((2, 3, 4), density=0.5)
assert s.coords.dtype == np.uint8
s = COO.from_numpy(np.zeros(1000))
assert s.coords.dtype == np.uint16
def test_empty_reduction():
x = np.zeros((2, 3, 4), dtype=np.float_)
xs = COO.from_numpy(x)
assert_eq(x.sum(axis=(0, 2)),
xs.sum(axis=(0, 2)))
def test_add_many_sparse_arrays():
x = COO({(1, 1): 1})
y = sum([x] * 100)
assert y.nnz < np.prod(y.shape)
def test_single_dimension():
x = COO([1, 3], [1.0, 3.0])
assert x.shape == (4,)
assert_eq(x, np.array([0, 1.0, 0, 3.0]))
def test_empty_shape():
x = COO(np.empty((0, 1), dtype=np.int8), [1.0])
assert x.shape == ()
assert ((2 * x).todense() == np.array(2.0)).all()
def test_reshape(a, b):
x = random_x(a)
s = COO.from_numpy(x)
assert_eq(x.reshape(b), s.reshape(b))
def test_slicing(index):
x = random_x((2, 3, 4))
s = COO.from_numpy(x)
assert_eq(x[index], s[index])
import pytest
import random
import numpy as np
import scipy.sparse
from sparse import COO
import sparse
x = np.zeros(shape=(2, 3, 4), dtype=np.float32)
for i in range(10):
x[random.randint(0, x.shape[0] - 1),
random.randint(0, x.shape[1] - 1),
random.randint(0, x.shape[2] - 1)] = random.randint(0, 100)
y = COO.from_numpy(x)
def random_x(shape, dtype=float):
x = np.zeros(shape=shape, dtype=float)
for i in range(max(5, np.prod(x.shape) // 10)):
x[tuple(random.randint(0, d - 1)
for d in x.shape)] = random.randint(0, 100)
return x
def assert_eq(x, y):
assert x.shape == y.shape
assert x.dtype == y.dtype
if hasattr(x, 'todense'):
xx = x.todense()
else:
def test_all_nan_reduction_warning(reduction, axis):
x = random_value_array(np.nan, 1.0)(2 * 3 * 4).reshape(2, 3, 4)
s = COO.from_numpy(x)
with pytest.warns(RuntimeWarning):
getattr(sparse, reduction)(s, axis=axis)
def calculate_third(atoms,
replicated_atoms,
third_order_delta,
distance_threshold=None,
is_verbose=False):
"""
Compute third order force constant matrices by using the central
difference formula for the approximation
"""
n_atoms = len(atoms.numbers)
replicated_atoms = replicated_atoms
n_replicas = int(replicated_atoms.positions.shape[0] / n_atoms)
i_at_sparse = []
i_coord_sparse = []
jat_sparse = []
j_coord_sparse = []
k_sparse = []
value_sparse = []
n_forces_to_calculate = n_replicas * (n_atoms * 3)**2
n_forces_done = 0
n_forces_skipped = 0
for iat in range(n_atoms):
for jat in range(n_replicas * n_atoms):
is_computing = True
m, j_small = np.unravel_index(jat, (n_replicas, n_atoms))
if (distance_threshold is not None):
dxij = atoms.positions[iat] - (list_of_replicas[m] +
atoms.positions[j_small])
if (np.linalg.norm(dxij) > distance_threshold):
is_computing = False
n_forces_skipped += 9
if is_computing:
if is_verbose:
logging.info('calculating forces on atoms: ' + str(iat) +
',' + str(jat))
for icoord in range(3):
for jcoord in range(3):
value = calculate_single_third(atoms, replicated_atoms,
iat, icoord, jat,
jcoord,
third_order_delta)
for id in range(value.shape[0]):
i_at_sparse.append(iat)
i_coord_sparse.append(icoord)
jat_sparse.append(jat)
j_coord_sparse.append(jcoord)
k_sparse.append(id)
value_sparse.append(value[id])
n_forces_done += 9
if (n_forces_done + n_forces_skipped % 300) == 0:
logging.info('Calculate third derivatives ' + str(
int((n_forces_done + n_forces_skipped) /
n_forces_to_calculate * 100)) + '%')
logging.info('total forces to calculate third : ' +
str(n_forces_to_calculate))
logging.info('forces calculated : ' + str(n_forces_done))
logging.info('forces skipped (outside distance threshold) : ' +
str(n_forces_skipped))
coords = np.array(
[i_at_sparse, i_coord_sparse, jat_sparse, j_coord_sparse, k_sparse])
shape = (n_atoms, 3, n_replicas * n_atoms, 3, n_replicas * n_atoms * 3)
phifull = COO(coords, np.array(value_sparse), shape)
phifull = phifull.reshape \
((n_atoms * 3, n_replicas * n_atoms * 3, n_replicas * n_atoms * 3))
return phifull
