def h_constraint_der_zero(grid_points):
"""
Boundary condition h'(0) = 0.
"""
h_coef = grid_points[N_COEF_U:]
h = Chebyshev(h_coef, domain=[0, radius])
return h.deriv(m=1)(0.0)
def __init__(self, n):
self.x0 = (1 + np.arange(n)) / (n + 1)
specs = []
super(ChebyshevQuadrature, self).__init__(n, n, specs)
cp = Chebyshev(1)
self.T_all = [cp.basis(i + 1, domain=[0.0, 1.0]) for i in range(n)]
def __init__(self, n):
self.x0 = (1 + np.arange(n)) / (n + 1)
specs = []
super(ChebyshevQuadrature, self).__init__(n, n, specs)
cp = Chebyshev(1)
self.T_all = [cp.basis(i + 1, domain=[0.0, 1.0]) for i in range(n)]
def background(self, two_theta, intensities):
n_deg = 5
spline = Chebyshev(coef=np.ones((5,)))
red_two_theta, red_intensities = self.remove_peaks(
two_theta, self.smooth_data(intensities))
# Fit the background using the reduced dataset
spline = spline.fit(red_two_theta, red_intensities, n_deg)
# Return the predicted background
background = np.array(spline(two_theta))
return background
def PolyDiff(self, u, x):
"""Differentiate data with polynomial interpolation.
This method takes a collection of points of size (2*self.cheb_width),
fits a Chebychev polynomial to the collection of points, and computes
the derivative at the central point. The derivatives are collated and
returned to the calling functions as a NumPy array.
Keyword arguments:
u -- values of some function
x -- corresponding x-coordinates where u is evaluated
Note: This throws out the data close to the edges since the polynomial
derivative only works well when we're looking at the middle of the
points fit.
"""
u = u.flatten()
x = x.flatten()
n = len(x)
# Initialize a numpy array for storing the derivative values
du = np.zeros((n - 2 * self.cheb_width, self.diff_order))
# Loop for fitting Cheb polynomials to each point
for j in range(self.cheb_width, n - self.cheb_width):
# Select the points on which to fit polynomial
points = np.arange(j - self.cheb_width, j + self.cheb_width)
# Fit the polynomial
poly = Cheb.fit(x[points], u[points], self.cheb_degree)
# Take derivatives
for d in range(1, self.diff_order + 1):
du[j - self.cheb_width, d - 1] = poly.deriv(m=d)(x[j])
return du
def polynomials(self) -> List[List[float]]:
"""The polynomials for the piecewise approximation.
Returns:
The polynomials for the piecewise approximation.
"""
if self.num_state_qubits is None:
return [[]]
# note this must be the private attribute since we handle missing breakpoints at
# 0 and 2 ^ num_qubits here (e.g. if the function we approximate is not defined at 0
# and the user takes that into account we just add an identity)
num_intervals = len(self._breakpoints)
# Calculate the polynomials
polynomials = []
for i in range(0, num_intervals - 1):
# Calculate the polynomial approximating the function on the current interval
poly = Chebyshev.interpolate(self._f_x, self._degree,
domain=[self._breakpoints[i],
self._breakpoints[i + 1]])
# Convert polynomial to the standard basis and rescale it for the rotation gates
poly = 2 * poly.convert(kind=np.polynomial.Polynomial).coef
# Convert to list and append
polynomials.append(poly.tolist())
# If the last breakpoint is < 2 ** num_qubits, add the identity polynomial
if self._breakpoints[-1] < 2 ** self.num_state_qubits:
polynomials = polynomials + [[2 * np.arcsin(1)]]
# If the first breakpoint is > 0, add the identity polynomial
if self._breakpoints[0] > 0:
polynomials = [[2 * np.arcsin(1)]] + polynomials
return polynomials
def u_constraint_zero(grid_points):
"""
Boundary condition u(0) = 0.
"""
u_coef = grid_points[:N_COEF_U]
u = Chebyshev(u_coef, domain=[0, radius])
return u(0.0)
def interpolate_cheb_from_col(cheb_column, bands, wave):
bluest = np.min(bands) #effective wavelength of bluest filter
reddest = np.max(bands) #effective wavelength of reddest filter
if len(cheb_column) != len(wave):
warnings.warn(
"wave must be an array with the same number of elements as cheb_column"
)
else:
funcs = {
n: Chebyshev(cheb_column[n], domain=[bluest, reddest])
for n in range(len(cheb_column))
}
polynom = np.zeros(len(cheb_column))
for i in range(len(cheb_column)):
if cheb_column[i][0] != -99:
polynom[i] = funcs[i](wave[i])
if wave[i] >= reddest or wave[i] <= bluest:
warnings.warn(
"The wavelength at which you're interpolating the Chebyshev polynomial for index:"
+ str(i) +
" is outside the wavelength range of the used bands")
else:
polynom[i] = -99
return polynom
def h_constraint(grid_points):
"""
Boundary condition h(r) = 0.
"""
h_coef = grid_points[N_COEF_U:]
h = Chebyshev(h_coef, domain=[0, radius])
return h(radius)
def _setup_approximants(self):
return [
Chebyshev.interpolate(self.function,
self.degree,
domain=window,
*self.args) for window in self._windows
]
def cheb_series(x):
sum = 0
alpha = np.sqrt(2) - 1
coef = np.zeros(11 * 2 + 1)
for i in range(11):
coef[2 * i + 1] = 1
sum += (-1)**i / (2 * i + 1) * alpha**(2 * i + 1) * Chebyshev(coef)(x)
coef[2 * i + 1] = 0
return 2 * sum
def get_mode_cheb_coeffs(self, mode_index, cheb_order):
"""
Cheb coeffs of a mode.
The projection process is performed on [0,1]^dim.
"""
import scipy.special as sps
cheby_nodes, _, cheby_weights = \
sps.chebyt(cheb_order).weights.T # pylint: disable=E1136,E0633
window = [0, 1]
cheby_nodes = cheby_nodes * (window[1] -
window[0]) / 2 + np.mean(window)
cheby_weights = cheby_weights * (window[1] - window[0]) / 2
mode = self.get_template_mode(mode_index)
grid = np.meshgrid(*[cheby_nodes for d in range(self.dim)],
indexing='ij')
mvals = mode(*grid)
from numpy.polynomial.chebyshev import Chebyshev
coef_scale = 2 * np.ones(cheb_order) / cheb_order
coef_scale[0] /= 2
basis_1d = np.array([
Chebyshev(coef=_orthonormal(cheb_order, i),
domain=window)(cheby_nodes) for i in range(cheb_order)
])
from itertools import product
if self.dim == 1:
basis_set = basis_1d
elif self.dim == 2:
basis_set = np.array([
b1.reshape([cheb_order, 1]) * b2.reshape([1, cheb_order])
for b1, b2 in product(*[basis_1d for d in range(self.dim)])
])
elif self.dim == 3:
basis_set = np.array([
b1.reshape([cheb_order, 1, 1]) *
b2.reshape([1, cheb_order, 1]) * b3.reshape([1, 1, cheb_order])
for b1, b2, b3 in product(*[basis_1d for d in range(self.dim)])
])
mode_cheb_coeffs = np.array([
np.sum(mvals * basis) for basis in basis_set
]) * _self_tp(coef_scale, self.dim).reshape(-1)
# purge small coeffs whose magnitude are less than 8 times machine epsilon
mode_cheb_coeffs[np.abs(mode_cheb_coeffs) < 8 *
np.finfo(mode_cheb_coeffs.dtype).eps] = 0
return mode_cheb_coeffs
def polynomials(self) -> List[List[float]]:
"""The polynomials for the piecewise approximation.
Returns:
The polynomials for the piecewise approximation.
Raises:
TypeError: If the input function is not in the correct format.
"""
if self.num_state_qubits is None:
return [[]]
# note this must be the private attribute since we handle missing breakpoints at
# 0 and 2 ^ num_qubits here (e.g. if the function we approximate is not defined at 0
# and the user takes that into account we just add an identity)
breakpoints = self._breakpoints
# Need to take into account the case in which no breakpoints were provided in first place
if breakpoints == [0]:
breakpoints = [0, 2**self.num_state_qubits]
num_intervals = len(breakpoints)
# Calculate the polynomials
polynomials = []
for i in range(0, num_intervals - 1):
# Calculate the polynomial approximating the function on the current interval
try:
# If the function is constant don't call Chebyshev (not necessary and gives errors)
if isinstance(self.f_x, (float, int)):
# Append directly to list of polynomials
polynomials.append([self.f_x])
else:
poly = Chebyshev.interpolate(
self.f_x, self.degree, domain=[breakpoints[i], breakpoints[i + 1]]
)
# Convert polynomial to the standard basis and rescale it for the rotation gates
poly = 2 * poly.convert(kind=np.polynomial.Polynomial).coef
# Convert to list and append
polynomials.append(poly.tolist())
except ValueError as err:
raise TypeError(
" <lambda>() missing 1 required positional argument: '"
+ self.f_x.__code__.co_varnames[0]
+ "'."
+ " Constant functions should be specified as 'f_x = constant'."
) from err
# If the last breakpoint is < 2 ** num_qubits, add the identity polynomial
if breakpoints[-1] < 2**self.num_state_qubits:
polynomials = polynomials + [[2 * np.arcsin(1)]]
# If the first breakpoint is > 0, add the identity polynomial
if breakpoints[0] > 0:
polynomials = [[2 * np.arcsin(1)]] + polynomials
return polynomials
def get_volume_from_h_coefs(h, radius):
"""
Returns volum of the bubble.
h is Chebyshev polynomial coefficients for shape.
radius is radius in A.
"""
r = Chebyshev([radius/2, radius/2], domain=[0, radius])
pol_to_integrate = h * r
return 2.0 * np.pi * (pol_to_integrate.integ()(radius) \
- pol_to_integrate.integ()(0.0))
def chebyshev(coefs, time, domain):
"""Evaluate a Chebyshev Polynomial.
Args:
coefs (list, np.array): Coefficients defining the polynomial
time (int, float): Time where to evaluate the polynomial
domain (list, tuple): Domain (or time interval) for which the polynomial is defined: [left, right]
Reference: Appendix A in the MSG Level 1.5 Image Data Format Description.
"""
return Chebyshev(coefs, domain=domain)(time) - 0.5 * coefs[0]
def refine_background(self, scattering_lengths, intensities, s=None, k=4):
"""Fit a univariate spline to the background data.
Arguments
---------
- scattering_lengths : Array of scattering vector lengths, q.
- intensities : Array of intensity values at each q position
- s : Smoothing factor passed to the spline. Default is
the variance of the background.
- k : Degree of the spline (default quartic spline).
"""
# Remove pre-indexed peaks for background fitting
phase_list = self.phases + self.background_phases
q, I = scattering_lengths, intensities
for phase in phase_list:
for reflection in phase.reflection_list:
q, I = remove_peak_from_df(x=q, y=I, xrange=reflection.qrange)
# Get an estimate for s from the non-peak data
if s is None:
s = np.std(I)
# Smoothing for background fitting
smoothI = savgol_filter(I, window_length=15, polyorder=5)
# Determine a background line from the noise without peaks
# self.spline = UnivariateSpline(
# x=q,
# y=I,
# s=s / 25,
# k=k,
# )
self.spline = Chebyshev(coef=np.ones((20,)))
self.spline = self.spline.fit(q, smoothI, 10)
# background = self.spline.cast(scattering_lengths)
# self.spline = splrep(q, I)
# Extrapolate the background for the whole pattern
# background = self.spline(scattering_lengths)
background = self.spline(scattering_lengths)
return background
def interpolate_cheb(cheb, bands, wave):
bluest = np.min(bands) #effective wavelength of bluest filter
reddest = np.max(bands) #effective wavelength of reddest filter
if wave[i] >= reddest or wave[i] <= bluest:
warnings.warn(
"The wavelength at which you're interpolating the Chebyshev polynomial is outside the wavelength range of the used bands"
)
func = Chebyshev(cheb,
domain=[bluest, reddest
]) # This evaluates the Chebyshev polynomial
return func(wave)
def get_free_energy_for_minimize(grid_points):
"""
Calculates total elastic energy for given radius and pressure.
"""
global free_energy_elastic_stretching, \
free_energy_elastic_bending, \
free_energy_elastic_tail, \
free_energy_external, \
current_volume
u_coef = grid_points[:N_COEF_U]
h_coef = grid_points[N_COEF_U:]
u = Chebyshev(u_coef, domain=[0, radius])
h = Chebyshev(h_coef, domain=[0, radius])
r = Chebyshev([radius/2, radius/2], domain=[0, radius])
dh_dr = h.deriv(m=1)
d2h_dr2 = h.deriv(m=2)
du_dr = u.deriv(m=1)
current_volume = get_volume_from_h_coefs(h, radius)
psi_str_to_integrate = \
(du_dr ** 2 + du_dr * dh_dr ** 2 \
+ 0.25 * dh_dr ** 4 \
+ (u // r) ** 2) * r
free_energy_elastic_stretching = np.pi * YOUNGS_MODULUS \
* (psi_str_to_integrate.integ()(radius) \
- psi_str_to_integrate.integ()(0.0))
free_energy_elastic_tail = np.pi * YOUNGS_MODULUS * u(radius) ** 2
psi_bend_to_integrate = \
(d2h_dr2 ** 2 + (dh_dr // r) ** 2) * r
free_energy_elastic_bending = RIGIDITY * np.pi \
* (psi_bend_to_integrate.integ()(radius) \
- psi_bend_to_integrate.integ()(0.0))
# free_energy_elastic_bending = 0.0
free_energy_external = -current_volume * pressure
return free_energy_elastic_stretching \
+ free_energy_elastic_bending \
+ free_energy_elastic_tail \
+ free_energy_external
def _normalize(spectra):
stars = spectra.index
wavelengths = spectra.flux.columns.values.copy()
flux = spectra.flux.values.copy()
error = spectra.error.reindex(columns=wavelengths).values.copy()
#TODO: Should negative fluxes be zero'd too?
bad_flux = sp.isnan(flux) | sp.isinf(flux)
bad_error = sp.isnan(error) | sp.isinf(error) | (error < 0)
bad = bad_flux | bad_error
flux[bad] = 1
error[bad] = ERROR_LIM
#TODO: Where does pixlist come from?
pixlist = sp.loadtxt('pixlist.txt', dtype=int)
var = sp.full_like(error, ERROR_LIM**2)
var[:, pixlist] = 0
inv_var = 1 / (var**2 + error**2)
norm_flux = sp.full_like(flux, 1)
norm_error = sp.full_like(error, ERROR_LIM)
for star in range(len(stars)):
for _, (left, right) in CHIPS.items():
mask = (left < wavelengths) & (wavelengths < right)
#TODO: Why are we using Chebyshev polynomials rather than smoothing splines?
#TODO: Why are we using three polynomials rather than one? Are spectra discontinuous between chips?
#TODO: Is the denominator being zero/negative ever an issue?
fit = Chebyshev.fit(x=wavelengths[mask],
y=flux[star][mask],
w=inv_var[star][mask],
deg=2)
norm_flux[star][mask] = flux[star][mask] / fit(wavelengths[mask])
norm_error[star][mask] = error[star][mask] / fit(wavelengths[mask])
#TODO: Why is the unreliability threshold different from the limit value?
unreliable = (norm_error > .3)
norm_flux[unreliable] = 1
norm_error[unreliable] = ERROR_LIM
# In the original, the masking is done in the parallax fitting code.
# Gonna do it earlier here to save a bit of memory.
mask = sp.any(
sp.vstack([(l < wavelengths) & (wavelengths < u)
for l, u in CHIPS.values()]), 0)
norm_flux = pd.DataFrame(norm_flux[:, mask], stars, wavelengths[mask])
norm_error = pd.DataFrame(norm_error[:, mask], stars, wavelengths[mask])
return pd.concat({'flux': norm_flux, 'error': norm_error}, 1)
def fit_func(f):
fn = 'fits/%s/fit%s.fits' % (f, f)
if os.path.exists(fn):
r = {}
d = pyfits.getdata('fits/%s/fit%s.fits' % (f, f), 'fit_info')[0]
ref = d.field('refwlband')
low = d.field('lowdwlband') + ref
high = d.field('highdwlband') + ref
d = pyfits.getdata('fits/%s/fit%s.fits' % (f, f), 'final_cheb')
for n in d.names:
r[n] = Chebyshev(d.field(n), (low, high))
else:
r = None
return r
def cheb_polys():
resolution = 513
X = np.linspace(-1, 1, resolution + 1)
n = 6
coeffs = np.zeros(n)
ax = plt.subplot(1, 1, 1)
for i in xrange(n):
coeffs[i] = 1
f = Chebyshev(coeffs)
plt.plot(X, f(X), label="$T_" + str(i) + "$")
coeffs[i] = 0
plt.ylim((-1.2, 1.2))
handles, labels = ax.get_legend_handles_labels()
ax.legend(handles, labels, loc="upper right")
plt.savefig("cheb_polys.pdf")
plt.clf()
def roots(self):
"""Utilises Boyd's O(n^2) recursive subdivision algorithm.
The chebfun
is recursively subsampled until it is successfully represented to
machine precision by a sequence of piecewise interpolants of degree
100 or less. A colleague matrix eigenvalue solve is then applied to
each of these pieces and the results are concatenated.
See:
J. P. Boyd, Computing zeros on a real interval through Chebyshev
expansion and polynomial rootfinding, SIAM J. Numer. Anal., 40
(2002), pp. 1666–1682.
"""
if self.size() == 1:
return np.array([])
elif self.size() <= 100:
ak = self.coefficients()
v = np.zeros_like(ak[:-1])
v[1] = 0.5
C1 = toeplitz(v)
C2 = np.zeros_like(C1)
C1[0, 1] = 1.
C2[-1, :] = ak[:-1]
C = C1 - .5 / ak[-1] * C2
eigenvalues = eigvals(C)
roots = [
eig.real for eig in eigenvalues if
np.allclose(eig.imag, 0, atol=1e-10) and np.abs(eig.real) <= 1
]
scaled_roots = self._ui_to_ab(np.array(roots))
return scaled_roots
else:
try:
# divide at a close-to-zero split-point
split_point = self._ui_to_ab(0.0123456789)
return np.concatenate(
(self.restrict([self._domain[0], split_point]).roots(),
self.restrict([split_point, self._domain[1]]).roots()))
except:
# Seems to have many fake roots for high degree fits
coeffs = self.coefficients()
domain = self._domain
possibilities = Chebyshev(coeffs, domain).roots()
return np.array(
[float(i.real) for i in possibilities if i.imag == 0.0])
def runge_plot():
a, b = -1, 1
resolution = 513
runge = lambda x: 1 / (1 + 25 * x**2)
X2 = np.linspace(a, b, resolution)
ax = plt.subplot(1, 1, 1)
plt.plot(X2, runge(X2), label="$\\frac{1}{1+25x^2}$")
for order in [5, 10, 15, 25]:
X = nodes(a, b, order)
Y = runge(X)
C = get_coefs(Y)
f = Chebyshev(C)
plt.plot(X2, f(X2), label="Order " + str(order))
handles, labels = ax.get_legend_handles_labels()
ax.legend(handles, labels, loc="upper right")
plt.savefig("runge_chebyshev.pdf")
plt.clf()
def cos_interp():
a = -1
b = 1
order = 20
resolution = 501
X = nodes(a, b, resolution)
F = lambda x: np.cos(x)
A = F(X)
cfs = get_coefs(A)
print "number of coefficients for cos that are greater than 1E-14: ", (
np.absolute(cfs) > 1E-14).sum()
f = Chebyshev(cfs)
X2 = np.linspace(a, b, resolution)
ax = plt.subplot(1, 1, 1)
plt.plot(X2, F(X2), label="$\\cos x$")
plt.plot(X2, f(X2), label="Chebyshev Interpolant")
handles, labels = ax.get_legend_handles_labels()
ax.legend(handles, labels, loc="upper right")
plt.show()
def crazy_interp():
# This one takes a little time to run
a = -1
b = 1
order = 100000
resolution = 100001
# Where to sample.
X = nodes(a, b, order)
F = lambda x: np.sin(1. / x) * np.sin(1. / np.sin(1. / x))
A = F(X)
cfs = get_coefs(A)
print "The last 10 coeffients are: ", cfs[-10:]
f = Chebyshev(cfs)
# Sample values for plot.
X2 = np.linspace(a, b, resolution)
ax = plt.subplot(1, 1, 1)
plt.plot(X2, f(X2), label="Chebyshev Interpolant")
plt.plot(X2,
F(X2),
label="$\\sin\\frac{1}{x} \\sin\\frac{1}{\\sin\\frac{1}{x}}$")
handles, labels = ax.get_legend_handles_labels()
ax.legend(handles, labels, loc="upper right")
plt.show()
def solve(
self,
d_mus:np.ndarray,
lambd=None,
):
n = self.n
k = self.k
if lambd is None:
lambd = np.zeros(self.k)
H = np.zeros(shape=(k, k))
fpoly = None
for i_step in range(self.n_steps):
if fpoly is None:
fvals = np.exp(lambd.dot(self.G))
cs = fit_cheby(fvals)
fpoly = Chebyshev(cs)
e_mu = np.array([
(fpoly*cb(i)).integ(lbnd=-1)(1)
for i in range(2*k)
])
grad = e_mu[:k] - d_mus
grad_norm = np.linalg.norm(grad)
grad_norm_old = grad_norm
Pval_old = e_mu[0] - lambd.dot(d_mus)
if self.verbose:
print("Step: {}, Grad: {}, P: {}".format(
i_step, grad_norm, Pval_old
))
if grad_norm < self.grad_tol:
break
for i in range(k):
for j in range(k):
H[i, j] = (e_mu[i+j] + e_mu[abs(i-j)]) / 2
step = -np.linalg.solve(H, grad)
dfdx = step.dot(grad)
stepScaleFactor = 1.0
newX = lambd + stepScaleFactor * step
alpha = .3
beta = .25
while True:
fvals = np.exp(newX.dot(self.G))
cs = fit_cheby(fvals)
fpoly = np.polynomial.chebyshev.Chebyshev(cs)
e_mu = np.array([
(fpoly * cb(i)).integ(lbnd=-1)(1)
for i in range(2 * k)
])
grad = e_mu[:k] - d_mus
grad_norm = np.linalg.norm(grad)
Pval_new = fpoly.integ(lbnd=-1)(1) - newX.dot(d_mus)
delta_change = Pval_old + alpha * stepScaleFactor * dfdx - Pval_new
# if (delta_change > -1e-6 and grad_norm < grad_norm_old) or stepScaleFactor < 1e-5:
if (delta_change > -1e-6 or stepScaleFactor < 1e-5):
break
else:
stepScaleFactor *= beta
if self.verbose:
print("step: {}, delta: {}".format(stepScaleFactor, delta_change))
newX = lambd + stepScaleFactor * step
lambd = newX
self.lambd = lambd
self.f_poly = self.get_fpoly(lambd)
self.cdf_poly = self.f_poly.integ(lbnd=-1)
def __init__(self, x0):
super(ChebyshevQuadrature, self).__init__(11, 11, 2.799761e-03, x0)
cp = Chebyshev(1)
self.T_all = [cp.basis(i, domain=[0.0, 1.0]) for i in range(11)]
def setup_numpy():
return Chebyshev.interpolate(np.sin, deg=10, domain=(0, 1))
def nextIter(self):
VNext = T.interpolate(lambda W: -self.negValue(W),
deg=self.order,
domain=[self.WMin, self.WMax])
return VNext
def guessSln(self):
self.V = T.interpolate(lambda W: self.u(W + self.y),
deg=self.order,
domain=[self.WMin, self.WMax])
def findPolicy(self):
self.policy = T.interpolate(lambda W: W + self.y -
(self.findOptW1(W) / self.R),
deg=self.order,
domain=[self.WMin, self.WMax])
def _derivative(c, m):
cheb = Chebyshev(c)
dcheb = cheb.deriv(m=m)
return dcheb.coef
