def __measure_stress(self, cell1, cell2, weight):
""" Measures stress from the polar decomposition of the transformation :math:`UPA = C`.
Args:
cell1 (ndarray): Lower basis.
cell2 (ndarray): Upper basis.
weight (float): Weight factor for common unit cell.
Returns:
float: :math:`\bar{\epsilon}_A + \bar{\epsilon}_ B`.
"""
from scipy.linalg import polar
A = cell1.copy()[:2, :2]
B = cell2.copy()[:2, :2]
C = A + weight * (B - A)
T1 = C @ np.linalg.inv(A)
T2 = C @ np.linalg.inv(B)
def measure(P):
eps = P - np.identity(2)
meps = np.sqrt((eps[0, 0]**2 + eps[1, 1]**2 +
eps[0, 0] * eps[1, 1] + eps[1, 0]**2) / 4)
return meps
U1, P1 = polar(T1) # this one goes counterclockwise
U2, P2 = polar(T2) # this one goes clockwise
# u is rotation, p is strain
meps1 = measure(P1)
meps2 = measure(P2)
stress = meps1 + meps2
return (stress, P1 - np.identity(2), P2 - np.identity(2))
def random_init(self, D):
"""Random initialization of the MPS and initializiation of the
environments.
"""
Lcanon = [
polar(rand(D * self.pdim, D) +
rand(D * self.pdim, D) * 1j)[0].reshape(D, self.pdim, D)
for i in range(self.cell_size)
]
Rcanon = [
polar(rand(D, self.pdim * D) +
rand(D, self.pdim * D) * 1j)[0].reshape(D, self.pdim, D)
for i in range(self.cell_size)
]
self.sites = Lcanon + Rcanon
self.c = rand(D, D) + rand(D, D) * 1j
self.c = self.c / norm(self.c)
self.center_bond = self.cell_size
# Begin of environment
LE = np.zeros((D, self.MPOdim, D))
LE[:, 0, :] = 1.
LE = LE / norm(LE)
self.LEnvironment[0] = LE
# End of environment
RE = np.zeros((D, self.MPOdim, D))
RE[:, -1, :] = 1.
RE = RE / norm(RE)
self.REnvironment[self.end_bond] = RE
def measure_stress(self) -> float:
"""Measures the stress on both unit cells."""
A = self.bottom.cell.copy()[:2, :2]
B = self.top.cell.copy()[:2, :2]
C = A + self._weight * (B - A)
T1 = C @ np.linalg.inv(A)
T2 = C @ np.linalg.inv(B)
def measure(P):
eps = P - np.identity(2)
meps = np.sqrt(
(
eps[0, 0] ** 2
+ eps[1, 1] ** 2
+ eps[0, 0] * eps[1, 1]
+ eps[1, 0] ** 2
)
/ 4
)
return meps
U1, P1 = polar(T1) # this one goes counterclockwise
U2, P2 = polar(T2) # this one goes clockwise
# u is rotation, p is strain
meps1 = measure(P1)
meps2 = measure(P2)
stress = meps1 + meps2
# return (stress, P1 - np.identity(2), P2 - np.identity(2))
return stress
def _find_matches(self) -> None:
"""
Finds and stores the ZSL matches
"""
self.zsl_matches = []
film_sg = SlabGenerator(
self.film_structure,
self.film_miller,
min_slab_size=1,
min_vacuum_size=3,
in_unit_planes=True,
center_slab=True,
primitive=True,
reorient_lattice=
False, # This is necessary to not screw up the lattice
)
sub_sg = SlabGenerator(
self.substrate_structure,
self.substrate_miller,
min_slab_size=1,
min_vacuum_size=3,
in_unit_planes=True,
center_slab=True,
primitive=True,
reorient_lattice=
False, # This is necessary to not screw up the lattice
)
film_slab = film_sg.get_slab(shift=0)
sub_slab = sub_sg.get_slab(shift=0)
film_vectors = film_slab.lattice.matrix
substrate_vectors = sub_slab.lattice.matrix
# Generate all possible interface matches
self.zsl_matches = list(
self.zslgen(film_vectors[:2], substrate_vectors[:2], lowest=False))
for match in self.zsl_matches:
xform = get_2d_transform(film_vectors, match.film_vectors)
strain, rot = polar(xform)
assert np.allclose(
strain, np.round(strain)
), "Film lattice vectors changed during ZSL match, check your ZSL Generator parameters"
xform = get_2d_transform(substrate_vectors,
match.substrate_vectors)
strain, rot = polar(xform)
assert np.allclose(
strain, strain.astype(int)
), "Substrate lattice vectors changed during ZSL match, check your ZSL Generator parameters"
def __optimize_cell_rotation(self, cell1, cell2, weight):
r""" Optimizes indivual cell rotation before matching the common unit cell.
Args:
cell1 (ndarray): Lower basis.
cell2 (ndarray): Upper basis.
weight (float): Weight factor for common unit cell.
Returns:
tuple: (:math:`\phi_A`, :math:`\phi_B`)
"""
from scipy.linalg import polar
from scipy.optimize import minimize
def measure(P):
eps = P - np.identity(2)
meps = np.sqrt((eps[0, 0]**2 + eps[1, 1]**2 +
eps[0, 0] * eps[1, 1] + eps[1, 0]**2) / 4)
return meps
def find(angle, start, target):
c = np.cos(angle)
s = np.sin(angle)
R = np.array([[c, -s], [s, c]])
newP = np.linalg.inv(R) @ target @ np.linalg.inv(start)
return measure(newP)
def f(params, args):
angle1, angle2 = params
A, B, C = args
m1 = find(angle1, A, C)
m2 = find(angle2, B, C)
return m1 + m2
A = cell1.copy()[:2, :2]
B = cell2.copy()[:2, :2]
C = A + weight * (B - A)
T1 = C @ np.linalg.inv(A)
T2 = C @ np.linalg.inv(B)
U1, P1 = polar(T1) # this one goes counterclockwise
U2, P2 = polar(T2) # this one goes clockwise
angle1 = np.arccos(U1[0, 0]) if U1[0, 0] < 0 else -np.arccos(U1[0, 0])
angle2 = np.arccos(U2[0, 0]) if U2[0, 0] < 0 else -np.arccos(U2[0, 0])
try:
res = minimize(f, x0=[angle1, angle2], args=[A, B, C])
newangle1, newangle2 = res.x
return (newangle1, newangle2)
except Exception as e:
print(e)
def polarDecompose(self, H):
from scipy.linalg import polar
T, L = self.separateTranslation(H) # T is translation matrix,
R, K = polar(L) # R is rotation matrix, K is stretch matrix
# The determinant of a rotation matrix must be positive
if np.linalg.det(R) < 0:
R[:3, :3] = -R[:3, :3]
K[:3, :3] = -K[:3, :3]
# Check answer still OK
assert np.allclose(L, R @ K), 'R*K should equal L, but it does not!'
assert np.allclose(H,
T @ R @ K), 'T*R*K should equal H, but it does not!'
# Decompose stretch matrix K into scale matrices
f, X = np.linalg.eig(
K) # eigenvalues and eigenvectors of stretch matrix
S = []
for factor, axis in zip(f[:3], X.T[:3]):
#if not np.isclose(factor, 1):
scale = np.eye(4) + np.outer(axis, axis) * (factor - 1)
S.append(scale)
# Check answers still OK
scale_prod = np.eye(4)
for scale in S:
scale_prod = scale_prod @ scale
if not np.allclose(K, scale_prod):
print(
'Product of scale matrices should equal stretch matrix K, but it does not!'
)
if not np.allclose(H, T @ R @ scale_prod):
print(
'T*R*(product of scale matrices) should equal stretch matrix K, but it does not!'
)
# Return all interesting outputs
return T, R, K, S, f, X
def testQdwhWithUpperTriangularInputAllOnes(self, m, n, log_cond):
"""Tests qdwh with upper triangular input of all ones."""
a = jnp.triu(jnp.ones((m, n))).astype(_QDWH_TEST_DTYPE)
u, s, v = jnp.linalg.svd(a, full_matrices=False)
cond = 10**log_cond
s = jnp.expand_dims(jnp.linspace(cond, 1, min(m, n)), range(u.ndim - 1))
a = (u * s) @ v
is_hermitian = _check_symmetry(a)
max_iterations = 10
actual_u, actual_h, _, _ = qdwh.qdwh(a, is_hermitian=is_hermitian,
max_iterations=max_iterations)
expected_u, expected_h = osp_linalg.polar(a)
# Sets the test tolerance.
rtol = 1E6 * _QDWH_TEST_EPS
with self.subTest('Test u.'):
relative_diff_u = _compute_relative_diff(actual_u, expected_u)
np.testing.assert_almost_equal(relative_diff_u, 1E-6, decimal=5)
with self.subTest('Test h.'):
relative_diff_h = _compute_relative_diff(actual_h, expected_h)
np.testing.assert_almost_equal(relative_diff_h, 1E-6, decimal=5)
with self.subTest('Test u.dot(h).'):
a_round_trip = _dot(actual_u, actual_h)
relative_diff_a = _compute_relative_diff(a_round_trip, a)
np.testing.assert_almost_equal(relative_diff_a, 1E-6, decimal=5)
with self.subTest('Test orthogonality.'):
actual_results = _dot(actual_u.T, actual_u)
expected_results = np.eye(n)
self.assertAllClose(
actual_results, expected_results, rtol=rtol, atol=1E-5)
def nearestPSD(A):
B = (A + A.T) / 2
H = sl.polar(B)[1]
return (B + H) / 2
def test_polarfunction(self):
'''Test routines to compute the matrix polar decomposition.'''
from scipy.linalg import polar
# Starting Matrix
matrix1 = self.create_matrix()
self.write_matrix(matrix1, self.input_file)
# Check Matrix
dense_check_u, dense_check_h = polar(matrix1.todense())
self.CheckMat = csr_matrix(dense_check_h)
# Result Matrix
input_matrix = nt.Matrix_ps(self.input_file, False)
u_matrix = nt.Matrix_ps(self.mat_dim)
h_matrix = nt.Matrix_ps(self.mat_dim)
permutation = nt.Permutation(input_matrix.GetLogicalDimension())
permutation.SetRandomPermutation()
self.isp.SetLoadBalance(permutation)
nt.SignSolvers.ComputePolarDecomposition(input_matrix, u_matrix,
h_matrix, self.isp)
h_matrix.WriteToMatrixMarket(result_file)
comm.barrier()
self.check_result()
comm.barrier()
self.CheckMat = csr_matrix(dense_check_u)
u_matrix.WriteToMatrixMarket(result_file)
self.check_result()
def randomized_expander(
z: np.ndarray,
q: np.ndarray,
n_col: int = 10,
n_iter: int = 2) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
# step A
m = z.shape[1]
p = min(n_col, m - q.shape[1])
w = npr.randn(m, p)
y = z @ w
del w
qy = partial_orthogonalization(y, q, overwrite_y=True)
q = np.append(q, qy, axis=1)
for _ in range(n_iter):
q = z @ (z.T @ q)
q = qr(q, mode='economic')[0]
# step B
h, c = qr(z.T @ q, mode='economic')
w, p = polar(c)
v, d = sym_eig(p)
return q @ v, d, (h @ w @ v).T
def testQdwhWithOnRankDeficientInput(self, m, n, log_cond):
"""Tests qdwh with rank-deficient input."""
a = jnp.triu(jnp.ones((m, n))).astype(_QDWH_TEST_DTYPE)
# Generates a rank-deficient input.
u, s, v = jnp.linalg.svd(a, full_matrices=False)
cond = 10**log_cond
s = jnp.linspace(cond, 1, min(m, n))
s = jnp.expand_dims(s.at[-1].set(0), range(u.ndim - 1))
a = (u * s) @ v
is_hermitian = _check_symmetry(a)
max_iterations = 15
actual_u, actual_h, _, _ = qdwh.qdwh(a, is_hermitian=is_hermitian,
max_iterations=max_iterations)
_, expected_h = osp_linalg.polar(a)
# Sets the test tolerance.
rtol = 1E4 * _QDWH_TEST_EPS
# For rank-deficient matrix, `u` is not unique.
with self.subTest('Test h.'):
relative_diff_h = _compute_relative_diff(actual_h, expected_h)
np.testing.assert_almost_equal(relative_diff_h, 1E-6, decimal=5)
with self.subTest('Test u.dot(h).'):
a_round_trip = _dot(actual_u, actual_h)
relative_diff_a = _compute_relative_diff(a_round_trip, a)
np.testing.assert_almost_equal(relative_diff_a, 1E-6, decimal=5)
with self.subTest('Test orthogonality.'):
actual_results = _dot(actual_u.T.conj(), actual_u)
expected_results = np.eye(n)
self.assertAllClose(
actual_results, expected_results, rtol=rtol, atol=1E-6)
def project_norm_pos_def(A):
"""
Calculates the nearest (in Frobenius norm) Symmetric Positive Definite matrix to A
https://www.sciencedirect.com/science/article/pii/0024379588902236
:param A: a square matrix
:return A_pd: the projection of A onto the space pf positive definite matrices
"""
assert A.ndim == 2 and A.shape[0] == A.shape[1], "A must be a square matrix"
# symmetrize A into B
B = (A + A.T) / 2
# Compute the symmetric polar factor H of B
_, H = polar(B)
A_pd = (B + H) / 2
# ensure symmetry
A_pd = (A_pd + A_pd.T) / 2
# test that A_pd is indeed PD. If not, then tweak it just a little bit
pd = False
k = 0
while not pd:
eig = np.linalg.eigvals(A_pd)
pd = np.all(eig > 0)
k += 1
if not pd:
mineig = min(eig)
A_pd = A_pd + (-mineig * k**2 + 10**-8) * np.eye(A.shape[0])
return A_pd
def fronebius_nearest_psd(A, return_distance=False):
"""Find the positive semi-definite matrix closest to `A`.
The closeness to `A` is measured by the Fronebius norm. The matrix closest to `A`
by that measure is uniquely defined in [3]_.
Parameters
----------
A : numpy.ndarray
Symmetric matrix
return_distance : bool, optional
Return distance of the input matrix to the approximation as given in
theorem 2.1 in [3]_.
This can be compared to the actual Frobenius norm between the
input and output to verify the calculation.
Returns
-------
X : numpy.ndarray
Positive semi-definite matrix approximating `A`.
Notes
-----
This function is a modification of [1]_, which is a Python adaption of [2]_, which
credits [3]_.
References
----------
.. [1] https://gist.github.com/fasiha/fdb5cec2054e6f1c6ae35476045a0bbd
.. [2] https://www.mathworks.com/matlabcentral/fileexchange/42885-nearestspd
.. [3] N.J. Higham, "Computing a nearest symmetric positive semidefinite
matrix" (1988): https://doi.org/10.1016/0024-3795(88)90223-6
"""
# pylint: disable=invalid-name
assert A.ndim == 2, "input is not a 2D matrix"
B = (A + A.T)/2.
_, H = lin.polar(B)
X = (B + H)/2.
# small numerical errors can make matrices that are not exactly
# symmetric, fix that
X = (X + X.T)/2.
# due to numerics, it's possible that the matrix is _still_ not psd.
# We can fix that iteratively by adding small increments of the identity matrix.
# This part comes from [1].
if not is_psd(X):
spacing = np.spacing(lin.norm(X))
I = np.eye(X.shape[0])
k = 1
while not is_psd(X):
mineig = np.min(np.real(lin.eigvals(X)))
X += I * (-mineig * k**2 + spacing)
k += 1
if return_distance:
C = (A - A.T)/2.
lam = lin.eigvalsh(B)
# pylint doesn't know that numpy.sum takes the "where" argument
# pylint: disable=unexpected-keyword-arg
dist = np.sqrt(np.sum(lam**2, where=lam < 0.) + lin.norm(C, ord='fro')**2)
return X, dist
return X
def polar_correct(list_a):
# We replace the matrices with the closest unitary matrices
# according to the Frobenius norm using polar decomposition
list_u = []
for a in list_a:
_u, _ = linalg.polar(a)
list_u.append(_u)
return list_u
def recoverM(A):
# Find the closest positive semi-definite matrix
# (via Higham)
B = 0.5 * (A + A.T)
H = polar(B)[1]
M = 0.5 * (B+H)
M = M / np.trace(M)
return M
def calc_deformation(src, dst):
A = affine_transform(src,dst)[:-1,:-1]
U,P = polar(A,side='left')
rotation = np.arctan2(U[1,0], U[0,0])
rotation = (rotation + np.pi) % (2 * np.pi )
return rotation, P
def reorient_bvecs(gtab, affines, atol=1e-2):
"""Reorient the directions in a GradientTable.
When correcting for motion, rotation of the diffusion-weighted volumes
might cause systematic bias in rotationally invariant measures, such as FA
and MD, and also cause characteristic biases in tractography, unless the
gradient directions are appropriately reoriented to compensate for this
effect [Leemans2009]_.
Parameters
----------
gtab : GradientTable
The nominal gradient table with which the data were acquired.
affines : list or ndarray of shape (n, 4, 4) or (n, 3, 3)
Each entry in this list or array contain either an affine
transformation (4,4) or a rotation matrix (3, 3).
In both cases, the transformations encode the rotation that was applied
to the image corresponding to one of the non-zero gradient directions
(ordered according to their order in `gtab.bvecs[~gtab.b0s_mask]`)
atol: see gradient_table()
Returns
-------
gtab : a GradientTable class instance with the reoriented directions
References
----------
.. [Leemans2009] The B-Matrix Must Be Rotated When Correcting for
Subject Motion in DTI Data. Leemans, A. and Jones, D.K. (2009).
MRM, 61: 1336-1349
"""
new_bvecs = gtab.bvecs[~gtab.b0s_mask]
if new_bvecs.shape[0] != len(affines):
e_s = "Number of affine transformations must match number of "
e_s += "non-zero gradients"
raise ValueError(e_s)
for i, aff in enumerate(affines):
if aff.shape == (4, 4):
# This must be an affine!
# Remove the translation component:
aff = aff[:3, :3]
# Decompose into rotation and scaling components:
R, S = polar(aff)
Rinv = inv(R)
# Apply the inverse of the rotation to the corresponding gradient
# direction:
new_bvecs[i] = np.dot(Rinv, new_bvecs[i])
return_bvecs = np.zeros(gtab.bvecs.shape)
return_bvecs[~gtab.b0s_mask] = new_bvecs
return gradient_table(gtab.bvals,
return_bvecs,
big_delta=gtab.big_delta,
small_delta=gtab.small_delta,
b0_threshold=gtab.b0_threshold,
atol=atol)
def compute_orientations(domain):
for key_block in domain.blocks:
block = domain.blocks[key_block]
for key_element in block.elements:
element = block.elements[key_element]
element.variables['R'] = dict()
element.variables['U'] = dict()
element.variables['orientation'] = dict()
for step in domain.times:
element.variables['R'][step], element.variables['U'][step] = \
polar(element.variables['F'][step])
element.variables['orientation'][step] = \
np.dot(
element.variables['R'][step],
np.dot(block.material.orientation, element.variables['R'][step].T))
for key_point in element.points:
point = element.points[key_point]
point.variables['R'] = dict()
point.variables['U'] = dict()
point.variables['orientation'] = dict()
for step in domain.times:
point.variables['R'][step], point.variables['U'][step] = \
polar(point.variables['F'][step])
point.variables['orientation'][step] = \
np.tensordot(
point.variables['R'][step],
np.tensordot(
block.material.orientation,
point.variables['R'][step].T,
axes = 1),
axes = 1)
def compute_orientations(domain):
for block in domain.blocks.values():
# block = domain.blocks[key_block]
for element in block.elements.values():
# element = block.elements[key_element]
element.variables['R'] = dict()
element.variables['U'] = dict()
element.variables['orientation'] = dict()
for step in domain.times:
element.variables['R'][step], element.variables['U'][step] = \
polar(element.variables['F'][step])
element.variables['orientation'][step] = \
np.dot(
element.variables['R'][step],
np.dot(block.material.orientation, element.variables['R'][step].T))
for point in element.points.values():
# point = element.points[key_point]
point.variables['R'] = dict()
point.variables['U'] = dict()
point.variables['orientation'] = dict()
for step in domain.times:
point.variables['R'][step], point.variables['U'][step] = \
polar(point.variables['F'][step])
point.variables['orientation'][step] = \
np.tensordot(
point.variables['R'][step],
np.tensordot(
block.material.orientation,
point.variables['R'][step].T,
axes = 1),
axes = 1)
def apply_gradient_component(self, wavepacket, component):
r"""Compute the effect of the gradient operator :math:`-i \varepsilon^2 \nabla_x`
on the basis functions :math:`\psi(x)` of a component :math:`\Phi_i` of the
new-kind Hagedorn wavepacket :math:`\Psi`.
:param wavepacket: The wavepacket :math:`\Psi` containing :math:`\Phi_i`.
:type wavepacket: A :py:class:`HagedornWavepacketBase` subclass instance.
:param component: The index :math:`i` of the component :math:`\Phi_i`.
:type component: Integer.
:return: Extended basis shape :math:`\mathfrak{\dot{K}}` and new coefficients :math:`c^\prime`
for component :math:`\Phi_i`. The coefficients are stored column-wise with
one column per dimension :math:`d`. The :math:`c^\prime` array is of shape
:math:`|\mathfrak{\dot{K}}| \times D`.
"""
D = wavepacket.get_dimension()
eps = wavepacket.get_eps()
q, p, Q, P, _ = wavepacket.get_parameters(component=component)
_, PA = polar(Q, side='left')
EW, EV = eigh(real(PA))
E = real(dot(P, inv(Q)))
F1 = dot(E, dot(inv(EV.T), diag(EW)))
F2 = 1.0j * dot(inv(EV.T), diag(1.0 / EW))
Gb = F1 - F2
Gf = F1 + F2
coeffs = wavepacket.get_coefficients(component=component)
# Prepare storage for new coefficients
K = wavepacket.get_basis_shapes(component=component)
Ke = K.extend()
size = Ke.get_basis_size()
cnew = zeros((size, D), dtype=complexfloating)
# We implement the more efficient scatter type stencil here
for k in K.get_node_iterator():
# Central phi_i coefficient
cnew[Ke[k], :] += squeeze(coeffs[K[k]] * p)
# Backward neighbours phi_{i - e_d}
nbw = Ke.get_neighbours(k, selection="backward")
for d, nb in nbw:
cnew[Ke[nb], :] += (sqrt(eps**2 / 2.0) *
sqrt(k[d]) * coeffs[K[k]] *
Gb[:, d])
# Forward neighbours phi_{i + e_d}
nfw = Ke.get_neighbours(k, selection="forward")
for d, nb in nfw:
cnew[Ke[nb], :] += (sqrt(eps**2 / 2.0) *
sqrt(k[d] + 1.0) * coeffs[K[k]] *
Gf[:, d])
return (Ke, cnew)
def dP(self, dF):
dR = polar(self.F + dF)[0] - self.R
# JFTinv = self.J*np.linalg.inv(self.F).transpose()
JFTinv = np.array([[self.F[1, 1], -self.F[1, 0]],
[-self.F[0, 1], self.F[0, 0]]])
dJFTinv = np.array([[dF[1, 1], -dF[1, 0]], [-dF[0, 1], dF[0, 0]]])
return 2 * self.mu * (dF - dR) + self.lambd * JFTinv * np.tensordot(
JFTinv, dF) + self.lambd * (self.J - 1) * dJFTinv
def checkdstable(A):
n = len(A)
P = solve_discrete_lyapunov(A.T, np.identity(n))
S = sqrtm(P)
invS = np.linalg.inv(S)
UB = S.dot(A).dot(invS)
[U, B] = polar(UB)
B = projectPSD(B, 0, 1)
return P, S, U, B
def get_list_noisy(list_goal_u, coeff, n):
list_u = []
for l in range(n):
error = coeff * (np.random.randn(n, n) + 1j * np.random.randn(n, n))
new_matrix = list_goal_u[l] + error
_u, _ = linalg.polar(new_matrix)
new_u = _u
list_u.append(new_u)
return list_u
def R(self):
"""
Return rotation part of ``DefGrad`` from polar decomposition.
"""
from scipy.linalg import polar
R, _ = polar(self)
return DefGrad(R)
def R(self):
"""
Return rotation part of ``DefGrad`` from polar decomposition.
"""
from scipy.linalg import polar
R, _ = polar(self)
return DefGrad(R)
def U(self):
"""
Return stretching part of ``DefGrad`` from right polar decomposition.
"""
from scipy.linalg import polar
_, U = polar(self, "right")
return DefGrad(U)
def V(self):
"""
Return stretching part of ``DefGrad`` from left polar decomposition.
"""
from scipy.linalg import polar
_, V = polar(self, "left")
return DefGrad(V)
def U(self):
"""
Return stretching part of ``DefGrad`` from right polar decomposition.
"""
from scipy.linalg import polar
_, U = polar(self, "right")
return DefGrad(U)
def V(self):
"""
Return stretching part of ``DefGrad`` from left polar decomposition.
"""
from scipy.linalg import polar
_, V = polar(self, "left")
return DefGrad(V)
def reorient_bvecs(gtab, affines):
"""Reorient the directions in a GradientTable.
When correcting for motion, rotation of the diffusion-weighted volumes
might cause systematic bias in rotationally invariant measures, such as FA
and MD, and also cause characteristic biases in tractography, unless the
gradient directions are appropriately reoriented to compensate for this
effect [Leemans2009]_.
Parameters
----------
gtab : GradientTable
The nominal gradient table with which the data were acquired.
affines : list or ndarray of shape (n, 4, 4) or (n, 3, 3)
Each entry in this list or array contain either an affine
transformation (4,4) or a rotation matrix (3, 3).
In both cases, the transformations encode the rotation that was applied
to the image corresponding to one of the non-zero gradient directions
(ordered according to their order in `gtab.bvecs[~gtab.b0s_mask]`)
Returns
-------
gtab : a GradientTable class instance with the reoriented directions
References
----------
.. [Leemans2009] The B-Matrix Must Be Rotated When Correcting for
Subject Motion in DTI Data. Leemans, A. and Jones, D.K. (2009).
MRM, 61: 1336-1349
"""
new_bvecs = gtab.bvecs[~gtab.b0s_mask]
if new_bvecs.shape[0] != len(affines):
e_s = "Number of affine transformations must match number of "
e_s += "non-zero gradients"
raise ValueError(e_s)
for i, aff in enumerate(affines):
if aff.shape == (4, 4):
# This must be an affine!
# Remove the translation component:
aff_no_trans = aff[:3, :3]
# Decompose into rotation and scaling components:
R, S = polar(aff_no_trans)
elif aff.shape == (3, 3):
# We assume this is a rotation matrix:
R = aff
Rinv = inv(R)
# Apply the inverse of the rotation to the corresponding gradient
# direction:
new_bvecs[i] = np.dot(Rinv, new_bvecs[i])
return_bvecs = np.zeros(gtab.bvecs.shape)
return_bvecs[~gtab.b0s_mask] = new_bvecs
return gradient_table(gtab.bvals, return_bvecs)
def _nifti_bvecs_to_worldspace_new(bvecs, meta):
from scipy.linalg import polar
from scipy.linalg import inv
R, S = polar(meta.frame)
for i in range(len(bvecs)):
bvecs[i] = np.dot(inv(R), bvecs[i])
#norm = la.norm(bvecs, axis=1)
#norm[norm == 0] = 1.0
#bvecs = bvecs / norm[:, None]
#bvecs[norm == 0] = np.array((0, 0, 1))
return bvecs
def get_noisy(list_u, coeff):
list_u_noisy = []
n = len(list_u[0])
m = len(list_u)
for l in range(m):
error = coeff * (np.random.randn(n, n) + 1j * np.random.randn(n, n))
new_matrix = list_u[l] + error
_u, _ = linalg.polar(new_matrix)
new_u = _u
list_u_noisy.append(new_u)
return list_u_noisy
def orthogonalize_cell(atoms: Atoms,
limit_denominator: int = 10,
return_strain=False):
"""
Make the cell of an ASE atoms object orthogonal. This is accomplished by repeating the cell until the x-component
of the lattice vectors in the xy-plane closely matches. If the ratio between the x-components is irrational this
may not be possible without introducing some strain. However, the amount of strain can be made arbitrarily small
by using many repetitions.
Parameters
----------
atoms : ASE atoms object
The non-orthogonal atoms object.
limit_denominator : int
The maximum denominator in the rational approximation. Increase this to allow more repetitions and hence less
strain.
return_strain : bool
If true, return the strain tensor that were applied to make the atoms orthogonal.
Returns
-------
atoms : ASE atoms object
The orthogonal atoms.
strain_tensor : 2x2 array
The applied strain tensor. Only provided if return_strain is true.
"""
if is_cell_orthogonal(atoms):
return atoms
atoms = atoms.copy()
atoms = standardize_cell(atoms)
fraction = atoms.cell[0, 0] / atoms.cell[1, 0]
fraction = Fraction(fraction).limit_denominator(limit_denominator)
atoms *= (fraction.denominator, fraction.numerator, 1)
new_cell = atoms.cell.copy()
new_cell[1, 0] = new_cell[0, 0]
a = np.linalg.solve(atoms.cell[:2, :2], new_cell[:2, :2])
_, strain_tensor = polar(a, side='left')
strain_tensor[0, 0] -= 1
strain_tensor[1, 1] -= 1
atoms.set_cell(new_cell, scale_atoms=True)
atoms.set_cell(np.diag(atoms.cell))
atoms.wrap()
if return_strain:
return atoms, strain_tensor
else:
return atoms
def execute(self, context):
global path, FilePath
Rigname = context.scene.RigNameLBS
if path[len(path) - len(Rigname) - 1:len(path) - 1] != Rigname:
path = FilePath + Rigname + '/'
Vrt = np.loadtxt(path + Rigname + '_vertz.txt', delimiter=',')
Fful = ReadTxt(path + Rigname + '_facz.txt')
NF = len(Fful)
NPs, NV = np.shape(Vrt)
NPs = NPs // 3
RM = np.zeros((NF, 9 * (NPs - 1)))
strttime = time.time()
t = 0
print('Computing Rotations....')
for f in Fful:
for ps in range(NPs):
if ps == 0:
RefFrm = Vrt[3 * ps:3 * ps + 3, f].T - np.mean(
Vrt[3 * ps:3 * ps + 3, f].T, axis=0)
nrm = np.cross(RefFrm[0], RefFrm[1]) / np.linalg.norm(
np.cross(RefFrm[0], RefFrm[1]))
RefFrm = np.concatenate((RefFrm, np.reshape(nrm, (1, 3))),
axis=0)
else:
DefFrm = Vrt[3 * ps:3 * ps + 3, f].T - np.mean(
Vrt[3 * ps:3 * ps + 3, f].T, axis=0)
nrm = np.cross(DefFrm[0], DefFrm[1]) / np.linalg.norm(
np.cross(DefFrm[0], DefFrm[1]))
DefFrm = np.concatenate((DefFrm, np.reshape(nrm, (1, 3))),
axis=0)
Q = np.dot(DefFrm.T, np.linalg.pinv(RefFrm.T))
R, S = scla.polar(Q)
RM[t, 9 * (ps - 1):9 * (ps - 1) + 9] = np.ravel(R)
t += 1
Ncl = context.scene.NumOfCls
print('Classifying....', Ncl, '...classes')
clustering = KMeans(n_clusters=Ncl).fit(RM)
Y = list(clustering.labels_)
Cls = [[]] * Ncl
t = 0
for i in list(Y):
Cls[i] = Cls[i] + [t]
t += 1
print("Segmentation time ...", time.time() - strttime)
WriteAsTxt(path + Rigname + "_ClusterKmeans.txt", Cls)
return {'FINISHED'}
def Us_from_A(A):
"""
N.B. I don't know how useful this will prove
Process to get the brick U1 and U2 from translationally invariant
single site A mps.
Multiple the two As together:
1) --A-A-- -> --B--
| | ||
Use QR decomp to turn B into:
2) -- C --D --
||
3) Reshape D into 4 x 4
Use Polar Decomp to turn D into a unitary:
| |
H
4) --D-- = | |
|| U_d
| |
5) Then multiply H into C and embed in a unitary
"""
# 1)
B = np.transpose(np.tensordot(A, A, axes=(2, 1)),
[1, 0, 3, 2]).reshape(2, 8)
#2)
C, D = qr(B, overwrite_a=True) # overwrite_a can give better performance
#3)
D = np.transpose(D.reshape(2, 2, 2, 2), [1, 2, 0, 3]).reshape(4, 4)
#4)
U_d, H = polar(D)
#5)
H = H.reshape(2, 2, 2, 2)
C_ = np.tensordot(H, C, axes=((2, 3), (1, 0))).reshape(4, 1)
C_ = C_ / np.linalg.norm(C_)
U_c = np.concatenate((C_, null_space(C_.conj().T)), axis=1)
return U_c, U_d
def verify_polar(a):
# Compute the polar decomposition, and then verify that
# the result has all the expected properties.
product_atol = np.sqrt(np.finfo(float).eps)
aa = np.asarray(a)
m, n = aa.shape
u, p = polar(a, side='right')
assert_equal(u.shape, (m, n))
assert_equal(p.shape, (n, n))
# a = up
assert_allclose(u.dot(p), a, atol=product_atol)
if m >= n:
assert_allclose(u.conj().T.dot(u), np.eye(n), atol=1e-15)
else:
assert_allclose(u.dot(u.conj().T), np.eye(m), atol=1e-15)
# p is Hermitian positive semidefinite.
assert_allclose(p.conj().T, p)
evals = eigh(p, eigvals_only=True)
nonzero_evals = evals[abs(evals) > 1e-14]
assert_((nonzero_evals >= 0).all())
u, p = polar(a, side='left')
assert_equal(u.shape, (m, n))
assert_equal(p.shape, (m, m))
# a = pu
assert_allclose(p.dot(u), a, atol=product_atol)
if m >= n:
assert_allclose(u.conj().T.dot(u), np.eye(n), atol=1e-15)
else:
assert_allclose(u.dot(u.conj().T), np.eye(m), atol=1e-15)
# p is Hermitian positive semidefinite.
assert_allclose(p.conj().T, p)
evals = eigh(p, eigvals_only=True)
nonzero_evals = evals[abs(evals) > 1e-14]
assert_((nonzero_evals >= 0).all())
def verify_polar(a):
# Compute the polar decomposition, and then verify that
# the result has all the expected properties.
product_atol = np.sqrt(np.finfo(float).eps)
aa = np.asarray(a)
m, n = aa.shape
u, p = polar(a, side='right')
assert_equal(u.shape, (m, n))
assert_equal(p.shape, (n, n))
# a = up
assert_allclose(u.dot(p), a, atol=product_atol)
if m >= n:
assert_allclose(u.conj().T.dot(u), np.eye(n), atol=1e-15)
else:
assert_allclose(u.dot(u.conj().T), np.eye(m), atol=1e-15)
# p is Hermitian positive semidefinite.
assert_allclose(p.conj().T, p)
evals = eigh(p, eigvals_only=True)
nonzero_evals = evals[abs(evals) > 1e-14]
assert_((nonzero_evals >= 0).all())
u, p = polar(a, side='left')
assert_equal(u.shape, (m, n))
assert_equal(p.shape, (m, m))
# a = pu
assert_allclose(p.dot(u), a, atol=product_atol)
if m >= n:
assert_allclose(u.conj().T.dot(u), np.eye(n), atol=1e-15)
else:
assert_allclose(u.dot(u.conj().T), np.eye(m), atol=1e-15)
# p is Hermitian positive semidefinite.
assert_allclose(p.conj().T, p)
evals = eigh(p, eigvals_only=True)
nonzero_evals = evals[abs(evals) > 1e-14]
assert_((nonzero_evals >= 0).all())
def project_to_unitary(parameters, check_unitary=True):
"""
Use polar decomposition to find the closest unitary matrix.
"""
# parameters must be a square number (TODO: FIRE)
n = int(np.sqrt(len(parameters)))
A = parameters.reshape(n, n)
U, p = polar(A, side='left')
if check_unitary:
# (np.conj is harmless for real U)
assert np.allclose(np.dot(U, np.conj(U.T)), np.eye(n))
parameters = U.reshape(n*n)
return parameters
def axisangle(self):
"""Return rotation part of ``DefGrad`` axis, angle tuple."""
from scipy.linalg import polar
R, _ = polar(self)
w, W = np.linalg.eig(R.T)
i = np.where(abs(np.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
axis = Vec3(np.real(W[:, i[-1]]).squeeze())
# rotation angle depending on direction
cosa = (np.trace(R) - 1.0) / 2.0
if abs(axis[2]) > 1e-8:
sina = (R[1, 0] + (cosa - 1.0) * axis[0] * axis[1]) / axis[2]
elif abs(axis[1]) > 1e-8:
sina = (R[0, 2] + (cosa - 1.0) * axis[0] * axis[2]) / axis[1]
else:
sina = (R[2, 1] + (cosa - 1.0) * axis[1] * axis[2]) / axis[0]
angle = np.rad2deg(np.arctan2(sina, cosa))
return axis, angle
def check_precomputed_polar(a, side, expected_u, expected_p):
# Compare the result of the polar decomposition to a
# precomputed result.
u, p = polar(a, side=side)
assert_allclose(u, expected_u, atol=1e-15)
assert_allclose(p, expected_p, atol=1e-15)
def psolve(a, b):
u, p = sla.polar(a)
return np.dot(np.dot(la.inv(p), la.inv(u)), b)
def polar_decomposition(self, side='right'):
"""
calculates matrices for polar decomposition
"""
return polar(self, side=side)
def _my_sqrtm_polar(self, X):
tol = 1e-10
L = np.linalg.cholesky(X + np.diag(np.ones(X.shape[0]) * tol))
U, P = polar(L, side="right")
return P
def recoverM(A):
B = 0.5 * (A + A.T)
H = polar(B)[1]
M = 0.5 * (B+H)
M = M / np.trace(M)
return M
def evaluate_basis_at(self, grid, component, *, prefactor=False):
r"""Evaluate the basis functions :math:`\phi_k` recursively at the given nodes :math:`\gamma`.
:param grid: The grid :math:`\Gamma` containing the nodes :math:`\gamma`.
:type grid: A class having a :py:meth:`get_nodes(...)` method.
:param component: The index :math:`i` of a single component :math:`\Phi_i` to evaluate.
:param prefactor: Whether to include a factor of :math:`\frac{1}{\sqrt{\det(Q)}}`.
:type prefactor: Boolean, default is ``False``.
:return: A two-dimensional ndarray :math:`H` of shape :math:`(|\mathfrak{K}_i|, |\Gamma|)` where
the entry :math:`H[\mu(k), i]` is the value of :math:`\phi_k(\gamma_i)`.
"""
D = self._dimension
bas = self._basis_shapes[component]
bs = self._basis_sizes[component]
# The grid
grid = self._grid_wrap(grid)
nodes = grid.get_nodes()
nn = grid.get_number_nodes(overall=True)
# Allocate the storage array
phi = zeros((bs, nn), dtype=complexfloating)
# Precompute some constants
Pi = self.get_parameters(component=component)
q, p, Q, P, _ = Pi
# Transformation to {w} basis
_, PA = polar(Q, side='left')
EW, EV = eigh(real(PA))
Qinv = dot(diag(1.0 / EW), EV.T)
QQ = identity(D)
# Compute the ground state phi_0 via direct evaluation
mu0 = bas[tuple(D * [0])]
phi[mu0, :] = self._evaluate_phi0(component, nodes, prefactor=False)
# Compute all higher order states phi_k via recursion
for d in range(D):
# Iterator for all valid index vectors k
indices = bas.get_node_iterator(mode="chain", direction=d)
for k in indices:
# Current index vector
ki = vstack(k)
# Access predecessors
phim = zeros((D, nn), dtype=complexfloating)
for j, kpj in bas.get_neighbours(k, selection="backward"):
mukpj = bas[kpj]
phim[j, :] = phi[mukpj, :]
# Compute 3-term recursion
p1 = (nodes - q) * phi[bas[k], :]
p2 = sqrt(ki) * phim
t1 = sqrt(2.0 / self._eps**2) * dot(Qinv[d, :], p1)
t2 = dot(QQ[d, :], p2)
# Find multi-index where to store the result
kped = bas.get_neighbours(k, selection="forward", direction=d)
# Did we find this k?
if len(kped) > 0:
kped = kped[0]
# Store computed value
phi[bas[kped[1]], :] = (t1 - t2) / sqrt(ki[d] + 1.0)
if prefactor is True:
phi = phi / self._get_sqrt(component)(det(Q))
return phi
def slim_recursion(self, grid, component, *, prefactor=False):
r"""Evaluate the Hagedorn wavepacket :math:`\Psi` at the given nodes :math:`\gamma`.
This routine is a slim version compared to the full basis evaluation. At every moment
we store only the data we really need to compute the next step until we hit the highest
order basis functions.
:param grid: The grid :math:`\Gamma` containing the nodes :math:`\gamma`.
:type grid: A class having a :py:meth:`get_nodes(...)` method.
:param component: The index :math:`i` of a single component :math:`\Phi_i` to evaluate.
:param prefactor: Whether to include a factor of :math:`\frac{1}{\sqrt{\det(Q)}}`.
:type prefactor: Boolean, default is ``False``.
:return: A list of arrays or a single array containing the values of the :math:`\Phi_i`
at the nodes :math:`\gamma`.
Note that this function does not include the global phase :math:`\exp(\frac{i S}{\varepsilon^2})`.
"""
D = self._dimension
# Precompute some constants
Pi = self.get_parameters(component=component)
q, p, Q, P, _ = Pi
# Transformation to {w} basis
_, PA = polar(Q, side='left')
EW, EV = eigh(real(PA))
Qinv = dot(diag(1.0 / EW), EV.T)
QQ = identity(D)
# The basis shape
bas = self._basis_shapes[component]
Z = tuple(D * [0])
# Book keeping
todo = []
newtodo = [Z]
olddelete = []
delete = []
tmp = {}
# The grid nodes
grid = self._grid_wrap(grid)
nn = grid.get_number_nodes(overall=True)
nodes = grid.get_nodes()
# Evaluate phi0
tmp[Z] = self._evaluate_phi0(component, nodes, prefactor=False)
psi = self._coefficients[component][bas[Z], 0] * tmp[Z]
# Iterate for higher order states
while len(newtodo) != 0:
# Delete results that never will be used again
for d in olddelete:
del tmp[d]
# Exchange queues
todo = newtodo
newtodo = []
olddelete = delete
delete = []
# Compute new results
for k in todo:
# Center stencil at node k
ki = vstack(k)
# Access predecessors
phim = zeros((D, nn), dtype=complexfloating)
for j, kpj in bas.get_neighbours(k, selection="backward"):
phim[j, :] = tmp[kpj]
# Compute the neighbours
for d, n in bas.get_neighbours(k, selection="forward"):
if n not in tmp.keys():
# Compute 3-term recursion
p1 = (nodes - q) * tmp[k]
p2 = sqrt(ki) * phim
t1 = sqrt(2.0 / self._eps**2) * dot(Qinv[d, :], p1)
t2 = dot(QQ[d, :], p2)
# Store computed value
tmp[n] = (t1 - t2) / sqrt(ki[d] + 1.0)
# And update the result
psi = psi + self._coefficients[component][bas[n], 0] * tmp[n]
newtodo.append(n)
delete.append(k)
if prefactor is True:
psi = psi / self._get_sqrt(component)(det(Q))
return psi
