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meanas/meanas/fdfd/bloch.py

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"""
Bloch eigenmode solver/operators
This module contains functions for generating and solving the
3D Bloch eigenproblem. The approach is to transform the problem
into the (spatial) fourier domain, transforming the equation
1/mu * curl(1/eps * curl(H_eigenmode)) = (w/c)^2 H_eigenmode
into
conv(1/mu_k, ik x conv(1/eps_k, ik x H_k)) = (w/c)^2 H_k
where:
- the `_k` subscript denotes a 3D fourier transformed field
- each component of `H_k` corresponds to a plane wave with wavevector `k`
- `x` is the cross product
- `conv()` denotes convolution
Since `k` and `H` are orthogonal for each plane wave, we can use each
`k` to create an orthogonal basis (k, m, n), with `k x m = n`, and
`|m| = |n| = 1`. The cross products are then simplified as follows:
- `h` is shorthand for `H_k`
- `(...)_xyz` denotes the `(x, y, z)` basis
- `(...)_kmn` denotes the `(k, m, n)` basis
- `hm` is the component of `h` in the `m` direction, etc.
We know
k @ h = kx hx + ky hy + kz hz = 0 = hk
h = hk + hm + hn = hm + hn
k = kk + km + kn = kk = |k|
We can write
k x h = (ky hz - kz hy,
kz hx - kx hz,
kx hy - ky hx)_xyz
= ((k x h) @ k, (k x h) @ m, (k x h) @ n)_kmn
= (0, (m x k) @ h, (n x k) @ h)_kmn # triple product ordering
= (0, kk (-n @ h), kk (m @ h))_kmn # (m x k) = -|k| n, etc.
= |k| (0, -h @ n, h @ m)_kmn
which gives us a straightforward way to perform the cross product
while simultaneously transforming into the `_kmn` basis.
We can also write
k x h = (km hn - kn hm,
kn hk - kk hn,
kk hm - km hk)_kmn
= (0, -kk hn, kk hm)_kmn
= (-kk hn)(mx, my, mz)_xyz + (kk hm)(nx, ny, nz)_xyz
= |k| (hm * (nx, ny, nz)_xyz
- hn * (mx, my, mz)_xyz)
which gives us a way to perform the cross product while simultaneously
trasnforming back into the `_xyz` basis.
We can also simplify `conv(X_k, Y_k)` as `fftn(X * ifftn(Y_k))`.
Using these results and storing `H_k` as `h = (hm, hn)`, we have
e_xyz = fftn(1/eps * ifftn(|k| (hm * n - hn * m)))
b_mn = |k| (-e_xyz @ n, e_xyz @ m)
h_mn = fftn(1/mu * ifftn(b_m * m + b_n * n))
which forms the operator from the left side of the equation.
We can then use a preconditioned block Rayleigh iteration algorithm, as in
SG Johnson and JD Joannopoulos, Block-iterative frequency-domain methods
for Maxwell's equations in a planewave basis, Optics Express 8, 3, 173-190 (2001)
(similar to that used in MPB) to find the eigenvectors for this operator.
===
Typically you will want to do something like
recip_lattice = numpy.diag(1/numpy.array(epsilon[0].shape * dx))
n, v = bloch.eigsolve(5, k0, recip_lattice, epsilon)
f = numpy.sqrt(-numpy.real(n[0]))
n_eff = norm(recip_lattice @ k0) / f
v2e = bloch.hmn_2_exyz(k0, recip_lattice, epsilon)
e_field = v2e(v[0])
k, f = find_k(frequency=1/1550,
tolerance=(1/1550 - 1/1551),
direction=[1, 0, 0],
G_matrix=recip_lattice,
epsilon=epsilon,
band=0)
"""
from typing import Callable, Any, cast, Sequence
import logging
import numpy
from numpy import pi, real, trace
from numpy.fft import fftfreq
from numpy.typing import NDArray, ArrayLike
import scipy # type: ignore
import scipy.optimize # type: ignore
from scipy.linalg import norm # type: ignore
import scipy.sparse.linalg as spalg # type: ignore
from ..fdmath import fdfield_t, cfdfield_t
logger = logging.getLogger(__name__)
try:
import pyfftw.interfaces.numpy_fft # type: ignore
import pyfftw.interfaces # type: ignore
import multiprocessing
logger.info('Using pyfftw')
pyfftw.interfaces.cache.enable()
pyfftw.interfaces.cache.set_keepalive_time(3600)
fftw_args = {
#'threads': multiprocessing.cpu_count(),
'overwrite_input': True,
#'planner_effort': 'FFTW_PATIENT',
}
def fftn(*args: Any, **kwargs: Any) -> NDArray[numpy.complex128]:
return pyfftw.interfaces.numpy_fft.fftn(*args, **kwargs, **fftw_args)
def ifftn(*args: Any, **kwargs: Any) -> NDArray[numpy.complex128]:
return pyfftw.interfaces.numpy_fft.ifftn(*args, **kwargs, **fftw_args)
except ImportError:
from numpy.fft import fftn, ifftn
logger.info('Using numpy fft')
def generate_kmn(
k0: ArrayLike,
G_matrix: ArrayLike,
shape: Sequence[int],
) -> tuple[NDArray[numpy.float64], NDArray[numpy.float64], NDArray[numpy.float64]]:
"""
Generate a (k, m, n) orthogonal basis for each k-vector in the simulation grid.
Args:
k0: [k0x, k0y, k0z], Bloch wavevector, in G basis.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
shape: [nx, ny, nz] shape of the simulation grid.
Returns:
`(|k|, m, n)` where `|k|` has shape `tuple(shape) + (1,)`
and `m`, `n` have shape `tuple(shape) + (3,)`.
All are given in the xyz basis (e.g. `|k|[0,0,0] = norm(G_matrix @ k0)`).
"""
k0 = numpy.array(k0)
G_matrix = numpy.array(G_matrix, copy=False)
Gi_grids = numpy.array(numpy.meshgrid(*(fftfreq(n, 1 / n) for n in shape[:3]), indexing='ij'))
Gi = numpy.moveaxis(Gi_grids, 0, -1)
k_G = k0[None, None, None, :] - Gi
k_xyz = numpy.moveaxis(G_matrix @ numpy.moveaxis(k_G, 3, 2), 3, 2)
m = numpy.broadcast_to([0, 1, 0], tuple(shape[:3]) + (3,)).astype(float)
n = numpy.broadcast_to([0, 0, 1], tuple(shape[:3]) + (3,)).astype(float)
xy_non0 = numpy.any(k_xyz[:, :, :, 0:1] != 0, axis=3)
if numpy.any(xy_non0):
u = numpy.cross(k_xyz[xy_non0], [0, 0, 1])
m[xy_non0, :] = u / norm(u, axis=1)[:, None]
z_non0 = numpy.any(k_xyz != 0, axis=3)
if numpy.any(z_non0):
v = numpy.cross(k_xyz[z_non0], m[z_non0])
n[z_non0, :] = v / norm(v, axis=1)[:, None]
k_mag = norm(k_xyz, axis=3)[:, :, :, None]
return k_mag, m, n
def maxwell_operator(
k0: ArrayLike,
G_matrix: ArrayLike,
epsilon: fdfield_t,
mu: fdfield_t | None = None
) -> Callable[[NDArray[numpy.complex128]], NDArray[numpy.complex128]]:
"""
Generate the Maxwell operator
conv(1/mu_k, ik x conv(1/eps_k, ik x ___))
which is the spatial-frequency-space representation of
1/mu * curl(1/eps * curl(___))
The operator is a function that acts on a vector h_mn of size `2 * epsilon[0].size`
See the `meanas.fdfd.bloch` docstring for more information.
Args:
k0: Bloch wavevector, `[k0x, k0y, k0z]`.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
mu: Magnetic permability distribution for the simulation.
Default None (1 everywhere).
Returns:
Function which applies the maxwell operator to h_mn.
"""
shape = epsilon[0].shape + (1,)
k_mag, m, n = generate_kmn(k0, G_matrix, shape)
epsilon = numpy.moveaxis(epsilon, 0, -1)
if mu is not None:
mu = numpy.moveaxis(mu, 0, -1)
def operator(h: NDArray[numpy.complex128]) -> NDArray[numpy.complex128]:
"""
Maxwell operator for Bloch eigenmode simulation.
h is complex 2-field in (m, n) basis, vectorized
Args:
h: Raveled h_mn; size `2 * epsilon[0].size`.
Altered in-place.
Returns:
Raveled conv(1/mu_k, ik x conv(1/eps_k, ik x h_mn)), returned
and overwritten in-place of `h`.
"""
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
#{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
# cross product and transform into xyz basis
d_xyz = (n * hin_m
- m * hin_n) * k_mag # noqa: E128
# divide by epsilon
temp = ifftn(d_xyz, axes=range(3)) # reuses d_xyz if using pyfftw
temp /= epsilon
e_xyz = fftn(temp, axes=range(3))
# cross product and transform into mn basis
b_m = numpy.sum(e_xyz * n, axis=3, keepdims=True) * -k_mag
b_n = numpy.sum(e_xyz * m, axis=3, keepdims=True) * +k_mag
if mu is None:
h_m, h_n = b_m, b_n
else:
# transform from mn to xyz
b_xyz = (m * b_m[:, :, :, None]
+ n * b_n[:, :, :, None]) # noqa: E128
# divide by mu
temp = ifftn(b_xyz, axes=range(3))
temp /= mu
h_xyz = fftn(temp, axes=range(3))
# transform back to mn
h_m = numpy.sum(h_xyz * m, axis=3)
h_n = numpy.sum(h_xyz * n, axis=3)
h.shape = (h.size,)
h = numpy.concatenate((h_m.ravel(), h_n.ravel()), axis=None, out=h) # ravel and merge
h.shape = (h.size, 1)
return h
return operator
def hmn_2_exyz(
k0: ArrayLike,
G_matrix: ArrayLike,
epsilon: fdfield_t,
) -> Callable[[NDArray[numpy.complex128]], cfdfield_t]:
"""
Generate an operator which converts a vectorized spatial-frequency-space
`h_mn` into an E-field distribution, i.e.
ifft(conv(1/eps_k, ik x h_mn))
The operator is a function that acts on a vector `h_mn` of size `2 * epsilon[0].size`.
See the `meanas.fdfd.bloch` docstring for more information.
Args:
k0: Bloch wavevector, `[k0x, k0y, k0z]`.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
Returns:
Function for converting `h_mn` into `E_xyz`
"""
shape = epsilon[0].shape + (1,)
epsilon = numpy.moveaxis(epsilon, 0, -1)
k_mag, m, n = generate_kmn(k0, G_matrix, shape)
def operator(h: NDArray[numpy.complex128]) -> cfdfield_t:
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
d_xyz = (n * hin_m
- m * hin_n) * k_mag # noqa: E128
# divide by epsilon
return numpy.array([ei for ei in numpy.moveaxis(ifftn(d_xyz, axes=range(3)) / epsilon, 3, 0)]) # TODO avoid copy
return operator
def hmn_2_hxyz(
k0: ArrayLike,
G_matrix: ArrayLike,
epsilon: fdfield_t
) -> Callable[[NDArray[numpy.complex128]], cfdfield_t]:
"""
Generate an operator which converts a vectorized spatial-frequency-space
`h_mn` into an H-field distribution, i.e.
ifft(h_mn)
The operator is a function that acts on a vector `h_mn` of size `2 * epsilon[0].size`.
See the `meanas.fdfd.bloch` docstring for more information.
Args:
k0: Bloch wavevector, `[k0x, k0y, k0z]`.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
Only `epsilon[0].shape` is used.
Returns:
Function for converting `h_mn` into `H_xyz`
"""
shape = epsilon[0].shape + (1,)
_k_mag, m, n = generate_kmn(k0, G_matrix, shape)
def operator(h: NDArray[numpy.complex128]) -> cfdfield_t:
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
h_xyz = (m * hin_m
+ n * hin_n) # noqa: E128
return numpy.array([ifftn(hi) for hi in numpy.moveaxis(h_xyz, 3, 0)])
return operator
def inverse_maxwell_operator_approx(
k0: ArrayLike,
G_matrix: ArrayLike,
epsilon: fdfield_t,
mu: fdfield_t | None = None,
) -> Callable[[NDArray[numpy.complex128]], NDArray[numpy.complex128]]:
"""
Generate an approximate inverse of the Maxwell operator,
ik x conv(eps_k, ik x conv(mu_k, ___))
which can be used to improve the speed of ARPACK in shift-invert mode.
See the `meanas.fdfd.bloch` docstring for more information.
Args:
k0: Bloch wavevector, `[k0x, k0y, k0z]`.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
mu: Magnetic permability distribution for the simulation.
Default None (1 everywhere).
Returns:
Function which applies the approximate inverse of the maxwell operator to `h_mn`.
"""
shape = epsilon[0].shape + (1,)
epsilon = numpy.moveaxis(epsilon, 0, -1)
if mu is not None:
mu = numpy.moveaxis(mu, 0, -1)
k_mag, m, n = generate_kmn(k0, G_matrix, shape)
def operator(h: NDArray[numpy.complex128]) -> NDArray[numpy.complex128]:
"""
Approximate inverse Maxwell operator for Bloch eigenmode simulation.
h is complex 2-field in (m, n) basis, vectorized
Args:
h: Raveled h_mn; size `2 * epsilon[0].size`.
Returns:
Raveled ik x conv(eps_k, ik x conv(mu_k, h_mn))
"""
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
#{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
if mu is None:
b_m, b_n = hin_m, hin_n
else:
# transform from mn to xyz
h_xyz = (m * hin_m[:, :, :, None]
+ n * hin_n[:, :, :, None]) # noqa: E128
# multiply by mu
temp = ifftn(h_xyz, axes=range(3))
temp *= mu
b_xyz = fftn(temp, axes=range(3))
# transform back to mn
b_m = numpy.sum(b_xyz * m, axis=3)
b_n = numpy.sum(b_xyz * n, axis=3)
# cross product and transform into xyz basis
e_xyz = (n * b_m
- m * b_n) / k_mag # noqa: E128
# multiply by epsilon
temp = ifftn(e_xyz, axes=range(3))
temp *= epsilon
d_xyz = fftn(temp, axes=range(3))
# cross product and transform into mn basis crossinv_t2c
h_m = numpy.sum(d_xyz * n, axis=3, keepdims=True) / +k_mag
h_n = numpy.sum(d_xyz * m, axis=3, keepdims=True) / -k_mag
h.shape = (h.size,)
h = numpy.concatenate((h_m, h_n), axis=None, out=h)
h.shape = (h.size, 1)
return h
return operator
def find_k(
frequency: float,
tolerance: float,
direction: ArrayLike,
G_matrix: ArrayLike,
epsilon: fdfield_t,
mu: fdfield_t | None = None,
band: int = 0,
k_bounds: tuple[float, float] = (0, 0.5),
k_guess: float | None = None,
solve_callback: Callable[..., None] | None = None,
iter_callback: Callable[..., None] | None = None,
v0: NDArray[numpy.complex128] | None = None,
) -> tuple[float, float, NDArray[numpy.complex128], NDArray[numpy.complex128]]:
"""
Search for a bloch vector that has a given frequency.
Args:
frequency: Target frequency.
tolerance: Target frequency tolerance.
direction: k-vector direction to search along.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
mu: Magnetic permability distribution for the simulation.
Default None (1 everywhere).
band: Which band to search in. Default 0 (lowest frequency).
k_bounds: Minimum and maximum values for k. Default (0, 0.5).
k_guess: Initial value for k.
solve_callback: TODO
iter_callback: TODO
Returns:
`(k, actual_frequency, eigenvalues, eigenvectors)`
The found k-vector and its frequency, along with all eigenvalues and eigenvectors.
"""
direction = numpy.array(direction) / norm(direction)
k_bounds = tuple(sorted(k_bounds)) # type: ignore # we know the length already...
assert len(k_bounds) == 2
if k_guess is None:
k_guess = sum(k_bounds) / 2
n = None
v = v0
def get_f(k0_mag: float, band: int = 0) -> float:
nonlocal n, v
k0 = direction * k0_mag # type: ignore
n, v = eigsolve(band + 1, k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu, y0=v, callback=iter_callback)
f = numpy.sqrt(numpy.abs(numpy.real(n[band])))
if solve_callback:
solve_callback(k0_mag, n, v, f)
return f
res = scipy.optimize.minimize_scalar(
lambda x: abs(get_f(x, band) - frequency),
k_guess,
method='Bounded',
bounds=k_bounds,
options={'xatol': abs(tolerance)},
)
assert n is not None
assert v is not None
return float(res.x * direction), float(res.fun + frequency), n, v
def eigsolve(
num_modes: int,
k0: ArrayLike,
G_matrix: ArrayLike,
epsilon: fdfield_t,
mu: fdfield_t | None = None,
tolerance: float = 1e-7,
max_iters: int = 10000,
reset_iters: int = 100,
y0: ArrayLike | None = None,
callback: Callable[..., None] | None = None,
) -> tuple[NDArray[numpy.complex128], NDArray[numpy.complex128]]:
"""
Find the first (lowest-frequency) num_modes eigenmodes with Bloch wavevector
k0 of the specified structure.
Args:
k0: Bloch wavevector, `[k0x, k0y, k0z]`.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
mu: Magnetic permability distribution for the simulation.
Default `None` (1 everywhere).
tolerance: Solver stops when fractional change in the objective
`trace(Z.H @ A @ Z @ inv(Z Z.H))` is smaller than the tolerance
max_iters: TODO
reset_iters: TODO
callback: TODO
y0: TODO, initial guess
Returns:
`(eigenvalues, eigenvectors)` where `eigenvalues[i]` corresponds to the
vector `eigenvectors[i, :]`
"""
k0 = numpy.array(k0, copy=False)
h_size = 2 * epsilon[0].size
kmag = norm(G_matrix @ k0)
#
# Generate the operators
#
mop = maxwell_operator(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
imop = inverse_maxwell_operator_approx(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
scipy_op = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=mop)
scipy_iop = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=imop)
y_shape = (h_size, num_modes)
prev_E = 0.0
d_scale = 1.0
prev_traceGtKG = 0.0
prev_theta = 0.5
D = numpy.zeros(shape=y_shape, dtype=complex)
Z: NDArray[numpy.complex128]
if y0 is None:
Z = numpy.random.rand(*y_shape) + 1j * numpy.random.rand(*y_shape)
else:
Z = numpy.array(y0, copy=False).T
while True:
Z *= num_modes / norm(Z)
ZtZ = Z.conj().T @ Z
try:
U = numpy.linalg.inv(ZtZ)
except numpy.linalg.LinAlgError:
Z = numpy.random.rand(*y_shape) + 1j * numpy.random.rand(*y_shape)
continue
trace_U = real(trace(U))
if trace_U > 1e8 * num_modes:
Z = Z @ scipy.linalg.sqrtm(U).astype(numpy.complex128).conj().T
prev_traceGtKG = 0
continue
break
Zt = numpy.empty(Z.shape[::-1], dtype=numpy.complex128)
AZ = numpy.empty(Z.shape, dtype=numpy.complex128)
for i in range(max_iters):
Zt = numpy.conj(Z.T, out=Zt)
ZtZ = Zt @ Z
U = numpy.linalg.inv(ZtZ)
AZ = scipy_op @ Z.copy()
ZtAZ = Zt @ AZ
ZtAZU = ZtAZ @ U
E_signed = real(trace(ZtAZU))
sgn = numpy.sign(E_signed)
E = numpy.abs(E_signed)
# G = AZU projected onto the space orthonormal to Z via (1 - ZUZt)
G = (AZ @ U - Z @ U @ ZtAZU) * sgn
if i > 0 and abs(E - prev_E) < tolerance * 0.5 * (E + prev_E + 1e-7):
logger.info(
'Optimization succeded: '
f'[change in trace] {abs(E - prev_E)} - 5e-8 '
f'< {tolerance} [tolerance] * {(E + prev_E) / 2} [value of trace]'
)
break
KG = scipy_iop @ G # Preconditioned steepest descent direction
traceGtKG = _rtrace_AtB(G, KG)
if prev_traceGtKG == 0 or i % reset_iters == 0:
logger.info('CG reset')
gamma = 0.0
else:
gamma = traceGtKG / prev_traceGtKG
prev_traceGtKG = traceGtKG
D = gamma / d_scale * D + KG
d_scale = num_modes / norm(D)
D *= d_scale
# Now know the direction (D), but need to find how far to go (alpha)
# We are still minimizing E = tr((Z + alpha D)t A (Z + alpha D) U')
# = tr(ZtAZU' + alpha DtAZU' + alpha ZtADU' + alpha**2 DtADU')
# = tr((ZtAZ + 2 alpha sym(DtAZ) + alpha**2 DtAD) U')
# = tr(R U')
# = tr(R Qi) = tr(R inv(Q))
# where
# R = ZtAZ + 2 alpha sym(DtAZ) + alpha**2 DtAD
#
# Q = (Z + alpha D)t (Z + alpha D)
# = inv(ZtZ + alpha ZtD + alpha DtZ + alpha**2 DtD)
#
# Qi = inv(Q) = U'
AD = scipy_op @ D.copy()
DtD = D.conj().T @ D
DtAD = D.conj().T @ AD
symZtD = _symmetrize(Zt @ D)
symZtAD = _symmetrize(Zt @ AD)
Qi_memo: list[float | None] = [None, None]
def Qi_func(theta: float) -> float:
nonlocal Qi_memo
if Qi_memo[0] == theta:
return cast(float, Qi_memo[1])
c = numpy.cos(theta)
s = numpy.sin(theta)
Q = c * c * ZtZ + s * s * DtD + 2 * s * c * symZtD
try:
Qi = numpy.linalg.inv(Q)
except numpy.linalg.LinAlgError:
logger.info('taylor Qi')
# if c or s small, taylor expand
if c < 1e-4 * s and c != 0:
DtDi = numpy.linalg.inv(DtD)
Qi = DtDi / (s * s) - 2 * c / (s * s * s) * (DtDi @ (DtDi @ symZtD).conj().T)
elif s < 1e-4 * c and s != 0:
ZtZi = numpy.linalg.inv(ZtZ)
Qi = ZtZi / (c * c) - 2 * s / (c * c * c) * (ZtZi @ (ZtZi @ symZtD).conj().T)
else:
raise Exception('Inexplicable singularity in trace_func')
Qi_memo[0] = theta
Qi_memo[1] = cast(float, Qi)
return cast(float, Qi)
def trace_func(theta: float) -> float:
c = numpy.cos(theta)
s = numpy.sin(theta)
Qi = Qi_func(theta)
R = c * c * ZtAZ + s * s * DtAD + 2 * s * c * symZtAD
trace = _rtrace_AtB(R, Qi)
return numpy.abs(trace)
if False:
def trace_deriv(theta):
Qi = Qi_func(theta)
c2 = numpy.cos(2 * theta)
s2 = numpy.sin(2 * theta)
F = -0.5*s2 * (ZtAZ - DtAD) + c2 * symZtAD
trace_deriv = _rtrace_AtB(Qi, F)
G = Qi @ F.conj().T @ Qi.conj().T
H = -0.5*s2 * (ZtZ - DtD) + c2 * symZtD
trace_deriv -= _rtrace_AtB(G, H)
trace_deriv *= 2
return trace_deriv * sgn
U_sZtD = U @ symZtD
dE = 2.0 * (_rtrace_AtB(U, symZtAD) -
_rtrace_AtB(ZtAZU, U_sZtD))
d2E = 2 * (_rtrace_AtB(U, DtAD) -
_rtrace_AtB(ZtAZU, U @ (DtD - 4 * symZtD @ U_sZtD)) -
4 * _rtrace_AtB(U, symZtAD @ U_sZtD))
# Newton-Raphson to find a root of the first derivative:
theta = -dE / d2E
if d2E < 0 or abs(theta) >= pi:
theta = -abs(prev_theta) * numpy.sign(dE)
# theta, new_E, new_dE = linmin(theta, E, dE, 0.1, min(tolerance, 1e-6), 1e-14, 0, -numpy.sign(dE) * K_PI, trace_func)
theta, n, _, new_E, _, _new_dE = scipy.optimize.line_search(
trace_func,
trace_deriv,
xk=theta,
pk=numpy.ones((1, 1)),
gfk=dE,
old_fval=E,
c1=min(tolerance, 1e-6),
c2=0.1,
amax=pi,
)
result = scipy.optimize.minimize_scalar(trace_func, bounds=(0, pi), tol=tolerance)
new_E = result.fun
theta = result.x
improvement = numpy.abs(E - new_E) * 2 / numpy.abs(E + new_E)
logger.info(f'linmin improvement {improvement}')
Z *= numpy.cos(theta)
Z += D * numpy.sin(theta)
prev_theta = theta
prev_E = E
if callback:
callback()
#
# Recover eigenvectors from Z
#
U = numpy.linalg.inv(ZtZ)
Y = Z @ scipy.linalg.sqrtm(U).astype(numpy.complex128)
W = Y.conj().T @ (scipy_op @ Y.copy())
eigvals, W_eigvecs = numpy.linalg.eig(W)
eigvecs = Y @ W_eigvecs
for i in range(len(eigvals)):
v = eigvecs[:, i]
n = eigvals[i]
v /= norm(v)
Av = (scipy_op @ v.copy())[:, 0]
eigness = norm(Av - (v.conj() @ Av) * v)
f = numpy.sqrt(-numpy.real(n))
df = numpy.sqrt(-numpy.real(n) + eigness)
neff_err = kmag * (1 / df - 1 / f)
logger.info(f'eigness {i}: {eigness}\n neff_err: {neff_err}')
order = numpy.argsort(numpy.abs(eigvals))
return eigvals[order], eigvecs.T[order]
'''
def linmin(x_guess, f0, df0, x_max, f_tol=0.1, df_tol=min(tolerance, 1e-6), x_tol=1e-14, x_min=0, linmin_func):
if df0 > 0:
x0, f0, df0 = linmin(-x_guess, f0, -df0, -x_max, f_tol, df_tol, x_tol, -x_min, lambda q, dq: -linmin_func(q, dq))
return -x0, f0, -df0
elif df0 == 0:
return 0, f0, df0
else:
x = x_guess
fx = f0
dfx = df0
isave = numpy.zeros((2,), numpy.intc)
dsave = numpy.zeros((13,), float)
x, fx, dfx, task = minpack2.dsrch(x, fx, dfx, f_tol, df_tol, x_tol, task,
x_min, x_max, isave, dsave)
for i in range(int(1e6)):
if task != 'F':
logging.info(f'search converged in {i} iterations')
break
fx = f(x, dfx)
x, fx, dfx, task = minpack2.dsrch(x, fx, dfx, f_tol, df_tol, x_tol, task,
x_min, x_max, isave, dsave)
return x, fx, dfx
'''
def _rtrace_AtB(
A: NDArray[numpy.complex128],
B: NDArray[numpy.complex128] | float,
) -> float:
return real(numpy.sum(A.conj() * B))
def _symmetrize(A: NDArray[numpy.complex128]) -> NDArray[numpy.complex128]:
return (A + A.conj().T) * 0.5
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