def finite_diff(self, values, order=1, acc=4):
"""
Compute the derivatives of values by finite differences.
Args:
values: array-like object with the values of the path.
order: Order of the derivative.
acc: Accuracy: 4 corresponds to a central difference with 5 points.
Returns:
ndarray with the derivative.
"""
values = np.asarray(values)
assert len(values) == len(self)
# Loop over the lines of the path, extract the data and
# differenciate f(s) where s is the distance along the line.
ders_on_lines = []
for line in self.lines:
vals_on_line = values[line]
h = self.ds[line[0]]
if not np.allclose(h, self.ds[line[:-1]]):
raise ValueError("For finite difference derivatives, the path must be homogeneous!\n" +
str(self.ds[line[:-1]]))
der = finite_diff(vals_on_line, h, order=order, acc=acc)
ders_on_lines.append(der)
return np.array(ders_on_lines)
def plot_der_potentials(self, ax=None, order=1, **kwargs):
"""
Plot the derivatives of vl and vloc potentials on axis ax.
Used to analyze the derivative discontinuity introduced by the RRKJ method at rc.
"""
ax, fig, plt = get_ax_fig_plt(ax)
from abipy.tools.derivatives import finite_diff
from scipy.interpolate import UnivariateSpline
lines, legends = [], []
for l, pot in self.potentials.items():
# Need linear mesh for finite_difference --> Spline input potentials on lin_rmesh
lin_rmesh, h = np.linspace(pot.rmesh[0], pot.rmesh[-1], num=len(pot.rmesh) * 4, retstep=True)
spline = UnivariateSpline(pot.rmesh, pot.values, s=0)
lin_values = spline(lin_rmesh)
vder = finite_diff(lin_values, h, order=order, acc=4)
line, = ax.plot(lin_rmesh, vder, **self._wf_pltopts(l, "ae"))
lines.append(line)
if l == -1:
legends.append("%s-order derivative Vloc" % order)
else:
legends.append("$s-order derivative PS l=%s" % str(l))
decorate_ax(ax, xlabel="r [Bohr]", ylabel="$D^%s \phi(r)$" % order,
title="Derivative of the ion Pseudopotentials",
lines=lines, legends=legends)
return fig
def finite_diff(self, values, order=1, acc=4):
"""
Compute the derivatives of values by finite differences.
Args:
values: array-like object with the values of the path.
order: Order of the derivative.
acc: Accuracy: 4 corresponds to a central difference with 5 points.
Returns:
ndarray with the derivative.
"""
values = np.asarray(values)
assert len(values) == len(self)
# Loop over the lines of the path, extract the data and
# differenciate f(s) where s is the distance along the line.
ders_on_lines = []
for line in self.lines:
vals_on_line = values[line]
h = self.ds[line[0]]
if not np.allclose(h, self.ds[line[:-1]]):
raise ValueError(
"For finite difference derivatives, the path must be homogeneous!\n"
+ str(self.ds[line[:-1]]))
der = finite_diff(vals_on_line, h, order=order, acc=acc)
ders_on_lines.append(der)
return np.array(ders_on_lines)
def test_poly(self):
"""Test derivatives of polynomials."""
x, h = np.linspace(0, 1, 800, retstep=True)
orders = [1, 2, 3, 4]
accuracies = [2, 4, 6]
f = x**4
dpolys = {
1: 4 * x**3,
2: 12 * x**2,
3: 24 * x,
4: 24 * np.ones(len(x)),
}
decs = {
1: 5,
2: 4,
3: 4,
4: 1,
}
for order in orders:
for acc in accuracies:
if order == 4 and acc == 6: continue
print("order %s, acc %s" % (order, acc))
yder = finite_diff(f, h, order=order, acc=acc)
print(np.max(np.abs(yder - dpolys[order])))
self.assert_almost_equal(yder,
dpolys[order],
decimal=decs[order])
def plot_der_potentials(self, ax=None, order=1, **kwargs):
"""
Plot the derivatives of vl and vloc potentials on axis ax.
Used to analyze the derivative discontinuity introduced by the RRKJ method at rc.
"""
ax, fig, plt = get_ax_fig_plt(ax)
from abipy.tools.derivatives import finite_diff
from scipy.interpolate import UnivariateSpline
lines, legends = [], []
for l, pot in self.potentials.items():
# Need linear mesh for finite_difference --> Spline input potentials on lin_rmesh
lin_rmesh, h = np.linspace(pot.rmesh[0], pot.rmesh[-1], num=len(pot.rmesh) * 4, retstep=True)
spline = UnivariateSpline(pot.rmesh, pot.values, s=0)
lin_values = spline(lin_rmesh)
vder = finite_diff(lin_values, h, order=order, acc=4)
line, = ax.plot(lin_rmesh, vder, **self._wf_pltopts(l, "ae"))
lines.append(line)
if l == -1:
legends.append("%s-order derivative Vloc" % order)
else:
legends.append("$s-order derivative PS l=%s" % str(l))
decorate_ax(ax, xlabel="r [Bohr]", ylabel="$D^%s \phi(r)$" % order,
title="Derivative of the ion Pseudopotentials",
lines=lines, legends=legends)
return fig
def plot_der_densities(self, ax=None, order=1, **kwargs):
"""
Plot the derivatives of the densitiers on axis ax.
Used to analyze possible derivative discontinuities
"""
ax, fig, plt = get_ax_fig_plt(ax)
from scipy.interpolate import UnivariateSpline
lines, legends = [], []
for name, rho in self.densities.items():
if name != "rhoM": continue
# Need linear mesh for finite_difference --> Spline input densities on lin_rmesh
lin_rmesh, h = np.linspace(rho.rmesh[0], rho.rmesh[-1], num=len(rho.rmesh) * 4, retstep=True)
spline = UnivariateSpline(rho.rmesh, rho.values, s=0)
lin_values = spline(lin_rmesh)
vder = finite_diff(lin_values, h, order=order, acc=4)
line, = ax.plot(lin_rmesh, vder) #, **self._wf_pltopts(l, "ae"))
lines.append(line)
legends.append("%s-order derivative of %s" % (order, name))
decorate_ax(ax, xlabel="r [Bohr]", ylabel="$D^%s \n(r)$" % order, title="Derivative of the charge densities",
lines=lines, legends=legends)
return fig
def test_poly(self):
"""Test derivatives of polynomials."""
x, h = np.linspace(0, 1, 800, retstep=True)
orders = [1,2,3,4]
accuracies = [2,4,6]
f = x**4
dpolys = {
1: 4 * x**3,
2: 12 * x**2,
3: 24 * x,
4: 24 * np.ones(len(x)),
}
decs = {
1: 5,
2: 4,
3: 4,
4: 1,
}
for order in orders:
for acc in accuracies:
if order == 4 and acc == 6: continue
print("order %s, acc %s" % (order, acc))
yder = finite_diff(f, h, order=order, acc=acc)
print(np.max(np.abs(yder - dpolys[order])))
self.assert_almost_equal(yder, dpolys[order], decimal=decs[order])
def plot_der_densities(self, ax=None, order=1, **kwargs):
"""
Plot the derivatives of the densitiers on axis ax.
Used to analyze possible derivative discontinuities
"""
ax, fig, plt = get_ax_fig_plt(ax)
from scipy.interpolate import UnivariateSpline
lines, legends = [], []
for name, rho in self.densities.items():
if name != "rhoM": continue
# Need linear mesh for finite_difference --> Spline input densities on lin_rmesh
lin_rmesh, h = np.linspace(rho.rmesh[0], rho.rmesh[-1], num=len(rho.rmesh) * 4, retstep=True)
spline = UnivariateSpline(rho.rmesh, rho.values, s=0)
lin_values = spline(lin_rmesh)
vder = finite_diff(lin_values, h, order=order, acc=4)
line, = ax.plot(lin_rmesh, vder) #, **self._wf_pltopts(l, "ae"))
lines.append(line)
legends.append("%s-order derivative of %s" % (order, name))
decorate_ax(ax, xlabel="r [Bohr]", ylabel="$D^%s \n(r)$" % order, title="Derivative of the charge densities",
lines=lines, legends=legends)
return fig
def get_fd_emass_d2(self, enes_kline, acc):
d2 = finite_diff(enes_kline,
self.dk,
order=2,
acc=acc,
index=self.kpos)
emass = 1. / (d2.value * (units.eV_to_Ha / units.bohr_to_ang**2))
return emass, d2
def finite_diff(self, order=1, acc=4):
"""
Compute the derivatives by finite differences.
Args:
order: Order of the derivative.
acc: Accuracy. 4 is fine in many cases.
Returns:
new :class:`Function1d` instance with the derivative.
"""
if self.h is None:
raise ValueError("Finite differences with inhomogeneous meshes are not supported")
return self.__class__(self.mesh, finite_diff(self.values, self.h, order=order, acc=acc))
def test_exp(self):
"""Test derivatives of exp(x)."""
x, h = np.linspace(0, 2, 800, retstep=True)
orders = [1,2,3,4]
accuracies = [2,4,6]
exp = np.exp(x)
decs = {
1: 4,
2: 4,
3: 4,
4: 2,
}
for order in orders:
for acc in accuracies:
if order == 4 and acc == 6: continue
#print("order %s, acc %s" % (order, acc))
yder = finite_diff(exp, h, order=order, acc=acc)
#print(np.max(np.abs(yder - exp)))
self.assert_almost_equal(yder, exp, decs[order])
d = finite_diff(exp, h, order=order, acc=acc, index=100)
assert yder[100] == d.value
def test_exp(self):
"""Test derivatives of exp(x)."""
x, h = np.linspace(0, 2, 800, retstep=True)
orders = [1,2,3,4]
accuracies = [2,4,6]
exp = np.exp(x)
decs = {
1: 4,
2: 4,
3: 4,
4: 2,
}
for order in orders:
for acc in accuracies:
if order == 4 and acc == 6: continue
print("order %s, acc %s" % (order, acc))
yder = finite_diff(exp, h, order=order, acc=acc)
print(np.max(np.abs(yder - exp)))
self.assert_almost_equal(yder, exp, decs[order])
def calculate_gruns_finite_differences(phfreqs, eig, iv0, volume, dv):
"""
Calculates the Gruneisen parameters from finite differences on the phonon frequencies.
Uses the eigenvectors to match the frequencies at different volumes.
Args:
phfreqs: numpy array with the phonon frequencies at different volumes. Shape (nvols, nqpts, 3*natoms)
eig: numpy array with the phonon eigenvectors at the different volumes. Shape (nvols, nqpts, 3*natoms, 3*natoms)
If None simple ordering will be used.
iv0: index of the 0 volume.
volume: volume of the structure at the central volume.
dv: volume variation.
Returns:
A numpy array with the gruneisen parameters. Shape (nqpts, 3*natoms)
"""
phfreqs = phfreqs.copy()
nvols = phfreqs.shape[0]
if eig is not None:
for i in range(nvols):
if i == iv0:
continue
for iq in range(phfreqs.shape[1]):
ind = match_eigenvectors(eig[iv0, iq], eig[i, iq])
phfreqs[i, iq] = phfreqs[i, iq][ind]
acc = nvols - 1
g = np.zeros_like(phfreqs[0])
for iq in range(phfreqs.shape[1]):
for im in range(phfreqs.shape[2]):
w = phfreqs[iv0, iq, im]
if w == 0:
g[iq, im] = 0
else:
g[iq, im] = -finite_diff(
phfreqs[:, iq, im], dv, order=1, acc=acc)[iv0] * volume / w
return g
def calculate_gruns_finite_differences(phfreqs, eig, iv0, volume, dv):
"""
Calculates the Gruneisen parameters from finite differences on the phonon frequencies. Uses the eigenvectors
to match the frequencies at different volumes.
Args:
phfreqs: numpy array with the phonon frequencies at different volumes. Shape (nvols, nqpts, 3*natoms)
eig: numpy array with the phonon eigenvectors at the different volumes. Shape (nvols, nqpts, 3*natoms, 3*natoms)
If None simple ordering will be used.
iv0: index of the 0 volume.
volume: volume of the structure at the central volume.
dv: volume variation.
Returns:
A numpy array with the gruneisen parameters. Shape (nqpts, 3*natoms)
"""
phfreqs = phfreqs.copy()
nvols = phfreqs.shape[0]
if eig is not None:
for i in range(nvols):
if i == iv0:
continue
for iq in range(phfreqs.shape[1]):
ind = match_eigenvectors(eig[iv0, iq], eig[i, iq])
phfreqs[i, iq] = phfreqs[i, iq][ind]
acc = nvols - 1
g = np.zeros_like(phfreqs[0])
for iq in range(phfreqs.shape[1]):
for im in range(phfreqs.shape[2]):
w = phfreqs[iv0, iq, im]
if w == 0:
g[iq, im] = 0
else:
g[iq, im] = - finite_diff(phfreqs[:, iq, im], dv, order=1, acc=acc)[iv0] * volume / w
return g
def test_complex(self):
"""complex functions are not supported"""
x, h = np.linspace(0, 1, 800, retstep=True)
cf = 1j * x
with self.assertRaises(ValueError):
return finite_diff(cf, h)
def test_complex(self):
"""complex functions are not supported"""
x, h = np.linspace(0, 1, 800, retstep=True)
cf = 1j*x
with self.assertRaises(ValueError):
return finite_diff(cf, h)
