def compute_recon_power_spectrum(fishcast,z,b=-1.,b2=-1.,bs=-1.,N=None):
'''
Returns the reconstructed power spectrum, following Stephen's paper.
'''
if b == -1.: b = compute_b(fishcast,z)
if b2 == -1: b2 = 8*(b-1)/21
if bs == -1: bs = -2*(b-1)/7
noise = 1/compute_n(fishcast,z)
if fishcast.experiment.HI: noise = castorinaPn(z)
if N is None: N = 1/compute_n(fishcast,z)
f = fishcast.cosmo.scale_independent_growth_factor_f(z)
bL1 = b-1.
bL2 = b2-8*(b-1)/21
bLs = bs+2*(b-1)/7
K,MU = fishcast.k,fishcast.mu
h = fishcast.params['h']
klin = np.logspace(np.log10(min(K)),np.log10(max(K)),fishcast.Nk)
mulin = MU.reshape((fishcast.Nk,fishcast.Nmu))[0,:]
plin = np.array([fishcast.cosmo.pk_cb_lin(k*h,z)*h**3. for k in klin])
zelda = Zeldovich_Recon(klin,plin,R=15,N=2000,jn=5)
kSparse,p0ktable,p2ktable,p4ktable = zelda.make_pltable(f,ngauss=3,kmin=min(K),kmax=max(K),nk=200,method='RecSym')
bias_factors = np.array([1, bL1, bL1**2, bL2, bL1*bL2, bL2**2, bLs, bL1*bLs, bL2*bLs, bLs**2,0,0,0])
p0Sparse = np.sum(p0ktable*bias_factors, axis=1)
p2Sparse = np.sum(p2ktable*bias_factors, axis=1)
p4Sparse = np.sum(p4ktable*bias_factors, axis=1)
p0,p2,p4 = Spline(kSparse,p0Sparse)(klin),Spline(kSparse,p2Sparse)(klin),Spline(kSparse,p4Sparse)(klin)
l0,l2,l4 = legendre(0),legendre(2),legendre(4)
Pk = lambda mu: p0*l0(mu) + p2*l2(mu) + p4*l4(mu)
result = np.array([Pk(mu) for mu in mulin]).T
return result.flatten() + N
def read_from_h5(file_name, **kwargs):
"""Read data from an H5 file in LVC format"""
import re
import h5py
from scipy.interpolate import InterpolatedUnivariateSpline as Spline
phase_re = re.compile('phase_l(?P<ell>.*)_m(?P<m>.*)')
amp_re = re.compile('amp_l(?P<ell>.*)_m(?P<m>.*)')
with h5py.File(file_name) as f:
t = f['NRtimes'][:]
ell_m = np.array([[int(match['ell']), int(match['m'])] for key in f for match in [phase_re.match(key)] if match])
ell_min = np.min(ell_m[:, 0])
ell_max = np.max(ell_m[:, 0])
data = np.empty((t.size, sf.LM_total_size(ell_min, ell_max)), dtype=complex)
for ell in range(ell_min, ell_max+1):
for m in range(-ell, ell+1):
amp = Spline(f['amp_l{0}_m{1}/X'.format(ell, m)][:],
f['amp_l{0}_m{1}/Y'.format(ell, m)][:],
k=int(f['amp_l{0}_m{1}/deg'.format(ell, m)][()]))(t)
phase = Spline(f['phase_l{0}_m{1}/X'.format(ell, m)][:],
f['phase_l{0}_m{1}/Y'.format(ell, m)][:],
k=int(f['phase_l{0}_m{1}/deg'.format(ell, m)][()]))(t)
data[:, sf.LM_index(ell, m, ell_min)] = amp * np.exp(1j * phase)
if 'auxiliary-info' in f and 'history.txt' in f['auxiliary-info']:
history = ("### " + f['auxiliary-info/history.txt'][()].decode().replace('\n', '\n### ')).split('\n')
else:
history = [""]
constructor_statement = "scri.LVC.read_from_h5('{0}')".format(file_name)
w = WaveformModes(t=t, data=data, ell_min=ell_min, ell_max=ell_max,
frameType=Inertial, dataType=h,
history=history, constructor_statement=constructor_statement,
r_is_scaled_out=True, m_is_scaled_out=True)
return w
def __init__(self):
super(RBBTable, self).__init__()
self.df = pd.read_hdf(
os.path.join(fu.codepath, 'data', 't_to_norm_rad.hdf'), 'df')
self.table_temps = self.df.index.values.astype('float')
self.t2rad = {}
self.rad2t = {}
# the radiances for abs(T) < 3 K are 0 for channels 3-5 which means that during the
# backward lookup of radiance to T, the 0 radiance can not be looked up functionally
# (it's now a relation and not a function anymore). This makes the Spline interpolator
# ignore the negative part which I cannot afford.
# The work-around is to interpolate from T -3 to 3 (which are impossibly close to 0
# anyway for channels 3-5), ignoring all 0 values for T in [-2..2]
sliced = self.df.ix[abs(self.df.index) > 2]
for ch in range(3, 10):
# store the Spline interpolators in dictionary, 1 per channel
self.t2rad[ch] = Spline(self.table_temps, self.df[ch], s=0.0, k=1)
# for channels 3-5, take the data without the values around 0:
if ch < 6:
data = sliced[ch]
temps = sliced.index.values.astype('float')
else:
data = self.df[ch]
temps = self.table_temps
# store the Spline interpolators in dictionary, 1 per channel
self.rad2t[ch] = Spline(data, temps, s=0.0, k=1)
def _setup_chi(self):
chival = self.provider.get_comoving_radial_distance(self.zarray)
zatchi = Spline(chival, self.zarray)
chiatz = Spline(self.zarray, chival)
chimin = np.min(chival) + 1.e-5
chimax = np.max(chival)
chival = np.linspace(chimin, chimax, self.Nchi)
zval = zatchi(chival)
chistar = \
self.provider.get_comoving_radial_distance(self.provider.get_param('zstar'))
chivalp = \
np.array(list(map(lambda x: np.linspace(x, chistar, self.Nchi_mag), chival)))
chivalp = chivalp.transpose()[0]
zvalp = zatchi(chivalp)
chi_result = {
'zatchi': zatchi,
'chiatz': chiatz,
'chival': chival,
'zval': zval,
'chivalp': chivalp,
'zvalp': zvalp
}
return chi_result
def iteration_step(den):
"""Performs one iteration of the Kohn-Sham cycle (see DFT cheat sheet)."""
# -- Step 1: build effective Kohn-Sham potential --
# determine w(r) via Poisson's equation for initial or mixed density
w, r = solve_poisson(den)
# find hom.sol. w_hom(r) = a*r to match BC w(r_max) = q_tot = z
beta = (z - w[-1]) / r[-1]
w += beta * r
print("[INFO] using w_hom(r) with beta = {:.4f}".format(beta))
# interpolate directly Hartree potential since we don't need w(r) anymore
# and include a Hartree-Fock-like exchange of -1/2 * Hartree potential
vh = Spline(r, 0.5 * w / r)
def veff(r):
"""Effective Kohn-Sham potential --> compare with DFT cheat sheet."""
vext = -z / r
return vh(r) + vext
# -- Step 2: solve single-particle Schrödinger-like equation --
# integration returns (u(r_i), r_i) --> u(r_0) = solver(E)[0][0]
en = newton(lambda en0: solve_rseq(en0, veff)[0][0], -2.0, maxiter=50)
# integrate again using the correct energy
u, r = solve_rseq(en, veff)
# normalize u^2 to 1 regardless of z-value since psi(r) = Y_00 * u(r) / r
norm = trapz(u ** 2, r)
u /= np.sqrt(norm)
print("[INFO] normalizing |u(r)|^2 from {:.3g} to 1".format(norm))
# -- Step 3: construct and interpolate new density --
den_new = Spline(r, z * u ** 2)
# -- Step 4: compute total energy
etot = energy_functional(den_new, vh, en)
return etot, den_new
def calc_sig_dist_spline(x,
y,
xref,
yref,
n_iter=5,
sig_clip=3,
plot=False,
smooth_fac=10000,
wx=None,
wy=None,
ret_bool=False):
#now sigma clip
sbool = np.ones(x.shape, dtype='bool')
for i in range(n_iter):
#import pdb; pdb.set_trace()
tx = Spline(x[sbool],
y[sbool],
xref[sbool] - x[sbool],
s=smooth_fac,
w=wx[sbool])
ty = Spline(x[sbool],
y[sbool],
yref[sbool] - y[sbool],
s=smooth_fac,
w=wy[sbool])
xout = tx.ev(x, y)
yout = ty.ev(x, y)
dx = xref - xout - x
dy = yref - yout - y
xcen = np.mean(dx[sbool])
xsig = np.std(dx[sbool])
ycen = np.mean(dy[sbool])
ysig = np.std(dy[sbool])
sbool_temp = (dx > xcen - sig_clip * xsig) * (
dx < xcen + sig_clip * xsig) * (dy > ycen - sig_clip * ysig) * (
dy < ycen + sig_clip * ysig)
if i != n_iter - 1:
sbool = sbool_temp * sbool
print 'trimmed ', len(sbool) - np.sum(sbool), ' stars'
if plot:
print 'number of residuals outside of -5,5', np.sum((dx > 5) +
(dx < -5) +
(dy < -5) +
(dy > 5))
plt.figure(35)
plt.subplot(121)
plt.hist(dx, bins=100, range=(-5, 5))
plt.title('X residual to fit')
plt.subplot(122)
plt.hist(dy, bins=100, range=(-5, 5))
plt.title('Y residual to fit')
plt.show()
if not ret_bool:
return tx, ty, dx, dy
else:
return tx, ty, dx, dy, sbool
def make_bao_plot(fname):
"""Does the work of making the BAO figure."""
zlist = [2.0, 3.0, 4.0, 5.0, 6.0]
clist = ['b', 'c', 'g', 'm', 'r']
# Now make the figure.
fig, ax = plt.subplots(1, 2, figsize=(6, 3.0), sharey=True)
ii, jj = 0, 0
for zz, col in zip(zlist, clist):
# Read the data from file.
aa = 1.0 / (1.0 + zz)
pkd = np.loadtxt(dpath + "HI_bias_{:06.4f}.txt".format(aa))[1:, :]
# Now read linear theory and put it on the same grid.
lin = np.loadtxt("../../data/pklin_{:06.4f}.txt".format(aa))
dk = pkd[1, 0] - pkd[0, 0]
kk = np.linspace(pkd[0, 0] - dk / 2, pkd[-1, 0] + dk / 2, 5000)
tmp = np.interp(kk, lin[:, 0], lin[:, 1])
lin = np.zeros_like(pkd)
for i in range(pkd.shape[0]):
lin[i, 0] = pkd[i, 0]
ww = np.nonzero((kk > pkd[i, 0] - dk / 2)
& (kk < pkd[i, 0] + dk / 2))
lin[i, 1] = np.sum(kk[ww]**2 * tmp[ww]) / np.sum(kk[ww]**2)
# Take out the broad band.
if False: # Use smoothing spline as broad-band/no-wiggle.
knots = np.arange(0.05, 0.5, 0.05)
ss = Spline(pkd[:, 0], pkd[:, 1], t=knots)
rat = pkd[:, 1] / ss(pkd[:, 0])
else: # Use Savitsky-Golay filter for no-wiggle.
ss = savgol_filter(pkd[:, 1], 7, polyorder=2)
rat = pkd[:, 1] / ss
ax[ii].plot(pkd[:,0],rat+0.2*(jj//2),col+'-',\
label="$z={:.1f}$".format(zz))
if False: # Use smoothing spline as broad-band/no-wiggle.
ss = Spline(pkd[:, 0], lin, t=knots)
rat = lin / ss(pkd[:, 0])
else: # Use Savitsky-Golay filter for no-wiggle.
ss = savgol_filter(lin[:, 1], 7, polyorder=2)
rat = lin[:, 1] / ss
ax[ii].plot(pkd[:, 0], rat + 0.2 * (jj // 2), col + ':')
ii = (ii + 1) % 2
jj = jj + 1
# Tidy up the plot.
for ii in range(ax.size):
ax[ii].legend(ncol=2, framealpha=0.5)
ax[ii].set_xlim(0.05, 0.4)
ax[ii].set_ylim(0.75, 1.5)
ax[ii].set_xscale('linear')
ax[ii].set_yscale('linear')
# Put on some more labels.
ax[0].set_xlabel(r'$k\quad [h\,{\rm Mpc}^{-1}]$')
ax[1].set_xlabel(r'$k\quad [h\,{\rm Mpc}^{-1}]$')
ax[0].set_ylabel(r'$P(k)/P_{\rm nw}(k)$+offset')
# and finish up.
plt.tight_layout()
plt.savefig(fname)
def make_calib_plot():
"""Does the work of making the calibration figure."""
# Now make the figure.
fig, ax = plt.subplots(1, 2, figsize=(6, 2.5))
# The left hand panel is DLA bias vs. redshift.
bDLA = np.loadtxt("boss_bDLA.txt")
ax[0].errorbar(bDLA[:, 0], bDLA[:, 1], yerr=bDLA[:, 2], fmt='o')
ax[0].fill_between([1.5,3.5],[1.99-0.11,1.99-0.11],\
[1.99+0.11,1.99+0.11],\
color='lightgrey',alpha=0.5)
# The N-body results.
bb = np.loadtxt("HI_bias_vs_z_fid.txt")
ax[0].plot(bb[:, 0], bb[:, 2], 'md')
ss = Spline(bb[::-1, 0], bb[::-1, 2])
ax[0].plot(np.linspace(1.5, 3.5, 100), ss(np.linspace(1.5, 3.5, 100)),
'm--')
#
ax[0].set_xlabel(r'$z$')
ax[0].set_ylabel(r'$b_{DLA}(z)$')
# Tidy up.
ax[0].set_xlim(1.95, 3.25)
ax[0].set_ylim(1, 3)
ax[0].set_xscale('linear')
ax[0].set_yscale('linear')
# The right hand panel is OmegaHI vs. z.
# Read in the data and convert to "normal" OmegaHI convention.
dd = np.loadtxt("omega_HI_obs.txt")
Ez = np.sqrt(0.3 * (1 + dd[:, 0])**3 + 0.7)
ax[1].errorbar(dd[:,0],1e-3*dd[:,1]/Ez**2,yerr=1e-3*dd[:,2]/Ez**2,\
fmt='s',mfc='None')
# Plot the fit line.
zz = np.linspace(0, 7, 100)
Ez = np.sqrt(0.3 * (1 + zz)**3 + 0.7)
ax[1].plot(zz, 4e-4 * (1 + zz)**0.6 / Ez**2, 'k-')
# Now plot the simulation points.
dd = np.loadtxt("omega_HI_sim.txt")
######################dd[:,1] *= 3e5*(1+(3.5/dd[:,0])**6) / 2e9
ax[1].plot(dd[:, 0], dd[:, 1], 'md')
ss = Spline(dd[:, 0], dd[:, 1])
ax[1].plot(np.linspace(2, 6, 100), ss(np.linspace(2, 6, 100)), 'm--')
# Tidy up the plot.
ax[1].set_xlim(1, 6.25)
ax[1].set_ylim(4e-6, 3e-4)
ax[1].set_xscale('linear')
ax[1].set_yscale('log')
# Put on some more labels.
ax[1].set_xlabel(r'$z$')
ax[1].set_ylabel(r'$\Omega_{HI}$')
# and finish up.
plt.tight_layout()
plt.savefig('calib.pdf')
def interpolate(self, fdata):
"""Interpolate from computed data to required data.
Parameters
----------
fdata : ndarray
Frequency-domain data corresponding to ``freq_compute``.
Returns
-------
full_data : ndarray
Frequency-domain data corresponding to ``freq_required``.
"""
# Pre-allocate result.
out = np.zeros(self.freq_required.size, dtype=np.complex128)
# 1. Interpolate between fmin and fmax.
# If freq_coarse is not exactly freq_required, we use cubic spline to
# interpolate from fmin to fmax.
if self.freq_coarse.size != self.freq_required.size:
int_real = Spline(np.log(self.freq_compute),
fdata.real)(np.log(self.freq_interpolate))
int_imag = Spline(np.log(self.freq_compute),
fdata.imag)(np.log(self.freq_interpolate))
out[self.ifreq_interpolate] = int_real + 1j * int_imag
# If they are the same, just fill in the data.
else:
out[self.ifreq_interpolate] = fdata
# 2. Extrapolate from freq_required.min to fmin using PCHIP.
# 2.a Extend freq_required/data by adding a point at 1e-100 Hz with
# - same real part as lowest computed frequency and
# - zero imaginary part.
freq_ext = np.r_[1e-100, self.freq_compute]
data_ext = np.r_[fdata[0].real - 1e-100j, fdata]
# 2.b Actual 'extrapolation' (now an interpolation).
ext_real = Pchip(freq_ext, data_ext.real)(self.freq_extrapolate)
ext_imag = Pchip(freq_ext, data_ext.imag)(self.freq_extrapolate)
out[self.ifreq_extrapolate] = ext_real + 1j * ext_imag
return out
def read_from_h5(file_name, **kwargs):
"""Read data from an H5 file in LVC format"""
import re
import h5py
from scipy.interpolate import InterpolatedUnivariateSpline as Spline
phase_re = re.compile("phase_l(?P<ell>.*)_m(?P<m>.*)")
amp_re = re.compile("amp_l(?P<ell>.*)_m(?P<m>.*)")
with h5py.File(file_name, "r") as f:
t = f["NRtimes"][:]
ell_m = np.array([[int(match["ell"]),
int(match["m"])] for key in f
for match in [phase_re.match(key)] if match])
ell_min = np.min(ell_m[:, 0])
ell_max = np.max(ell_m[:, 0])
data = np.empty((t.size, sf.LM_total_size(ell_min, ell_max)),
dtype=complex)
for ell in range(ell_min, ell_max + 1):
for m in range(-ell, ell + 1):
amp = Spline(f[f"amp_l{ell}_m{m}/X"][:],
f[f"amp_l{ell}_m{m}/Y"][:],
k=int(f[f"amp_l{ell}_m{m}/deg"][()]))(t)
phase = Spline(f[f"phase_l{ell}_m{m}/X"][:],
f[f"phase_l{ell}_m{m}/Y"][:],
k=int(f[f"phase_l{ell}_m{m}/deg"][()]))(t)
data[:,
sf.LM_index(ell, m, ell_min)] = amp * np.exp(1j * phase)
if "auxiliary-info" in f and "history.txt" in f["auxiliary-info"]:
history = ("### " +
f["auxiliary-info/history.txt"][()].decode().replace(
"\n", "\n### ")).split("\n")
else:
history = [""]
constructor_statement = f"scri.LVC.read_from_h5('{file_name}')"
w = WaveformModes(
t=t,
data=data,
ell_min=ell_min,
ell_max=ell_max,
frameType=Inertial,
dataType=h,
history=history,
constructor_statement=constructor_statement,
r_is_scaled_out=True,
m_is_scaled_out=True,
)
return w
def make_bao_plot():
"""Does the work of making the BAO figure."""
zlist = [2.0, 3.0, 4.0, 5.0, 6.0]
clist = ['b', 'c', 'g', 'm', 'r']
# Now make the figure.
fig, ax = plt.subplots(1, 2, figsize=(6, 3.0), sharey=True)
ii, jj = 0, 0
for zz, col in zip(zlist, clist):
# Read the data from file.
aa = 1.0 / (1.0 + zz)
pkd = np.loadtxt("HI_pks_1d_{:06.4f}.txt".format(aa))
# Now read linear theory and put it on the same grid -- currently
# not accounting for finite bin width.
lin = np.loadtxt("pklin_{:06.4f}.txt".format(aa))
lin = np.interp(pkd[:, 0], lin[:, 0], lin[:, 1])
# Take out the broad band.
if False: # Use smoothing spline as broad-band/no-wiggle.
knots = np.arange(0.05, 0.5, 0.05)
ss = Spline(pkd[:, 0], pkd[:, 1], t=knots)
rat = pkd[:, 1] / ss(pkd[:, 0])
else: # Use Savitsky-Golay filter for no-wiggle.
ss = savgol_filter(pkd[:, 1], 7, polyorder=2)
rat = pkd[:, 1] / ss
ax[ii].plot(pkd[:,0],rat+0.2*(jj//2),col+'-',\
label="$z={:.1f}$".format(zz))
if False: # Use smoothing spline as broad-band/no-wiggle.
ss = Spline(pkd[:, 0], lin, t=knots)
rat = lin / ss(pkd[:, 0])
else: # Use Savitsky-Golay filter for no-wiggle.
ss = savgol_filter(lin, 7, polyorder=2)
rat = lin / ss
ax[ii].plot(pkd[:, 0], rat + 0.2 * (jj // 2), col + ':')
ii = (ii + 1) % 2
jj = jj + 1
# Tidy up the plot.
for ii in range(ax.size):
ax[ii].legend(ncol=2, framealpha=0.5)
ax[ii].set_xlim(0.05, 0.4)
ax[ii].set_ylim(0.75, 1.5)
ax[ii].set_xscale('linear')
ax[ii].set_yscale('linear')
# Put on some more labels.
ax[0].set_xlabel(r'$k\quad [h\,{\rm Mpc}^{-1}]$')
ax[1].set_xlabel(r'$k\quad [h\,{\rm Mpc}^{-1}]$')
ax[0].set_ylabel(r'$P(k)/P_{\rm nw}(k)$+offset')
# and finish up.
plt.tight_layout()
plt.savefig('HI_bao.pdf')
def smooth_function_from_coordinates(
x,
y, # coordinates of some curve
sample_fraction=1.0, # fraction of x,y points used for resampling
spline_smoothing=4, # degree of spline (1 gives piecewise linear)
):
"""
Given a set of coordinates in the `x` and `y` arrays, create a
smooth function from these coordinates by 1) resampling n
uniformly distributed points by linear interpolation of the `x`
and `y` coordinates, where n is given by `sample_fraction` times
the length of `x`; and 2) interpolating the resampled points by
a smooth spline, where `spline_smoothing` is an integer holding
the degree of the piecewise polynomial pieces of the spline
(0 and 1 gives a piecewise linear function, 2 and higher gives
splines of that order). Return the smooth function as a
Python function of x, together with the (uniformly distributed)
resampled points on which the smooth function is based.
"""
# Construct linear interpolator of data points
from Scientific.Functions.Interpolation \
import InterpolatingFunction
linear = InterpolatingFunction([x], y)
# Resample
xp = np.linspace(x[0], x[-1], sample_fraction * len(x))
yp = np.array([linear(xi) for xi in xp])
# Spline smoothing or linear interpolation, based on (xp,yp)
if spline_smoothing >= 2:
from scipy.interpolate import UnivariateSpline as Spline
function = Spline(xp, yp, s=0, k=spline_smoothing)
else:
function = InterpolatingFunction([xp], yp)
return function, xp, yp
def main():
print("\n/---------------------------------------------------\\")
print("| Ass.5.2: Self-consistent loop for the helium atom |")
print("\\---------------------------------------------------/\n")
print("[INIT] parameter settings for this run:")
print(" electrons: {}".format(z))
print(" step size h: {}".format(h))
print(" minimum r: {}".format(tmin))
print(" maximum r: {}".format(tmax))
print(" grid pts: {}".format(nsteps))
print(" dense grid pts: {}".format(nsteps_dense))
print(" max. iterations: {}".format(itermax))
print(" convergence dE: {}".format(etol))
print(" mixing alpha: {}".format(amix))
print(" initial density: {}".format(init_density))
# -- Step 0: initialize density with exact H-atom-like density --
if init_density == "H-atom":
den_init = lambda r: z * z ** 3 * r * np.exp(-2 * z * r) # noqa
# -- alternatively initialize density with random numbers --
elif init_density == "random":
den_init = lambda r: random.randrange(0, 2) # noqa
else:
raise ValueError("unknown initial density setting")
# init loop variables
den_mix = den_old = den_init
etot_old = 0
iterstep = 0
# Kohn-Sham cycle: repeat until convergence is achieved
while True:
iterstep += 1
print("\n\n[ITER] starting {}. iteration step".format(iterstep))
# -- Steps 1 to 4: solve Kohn-Sham equations and find total energy --
etot_new, den_new = iteration_step(den_mix)
ediff = abs(etot_new - etot_old)
print("[INFO] energy difference dE = {:.3e} Ha".format(ediff))
# -- Step 5: check for convergence --
if iterstep >= itermax and ediff > etol:
print("\n[STOP] could not achieve convergence in 20 iterations\n")
break
elif ediff > etol:
# -- Step 6: mix densities for faster convergence --
r = np.linspace(tmin, tmax, nsteps_dense)
den_mix = Spline(r, amix * den_new(r) + (1 - amix) * den_old(r))
den_old = den_new
etot_old = etot_new
else:
print("\n[STOP] convergence achieved in", iterstep, "steps\n")
print("/------------------------------------\\")
print("| >>> final total energy <<< |")
print("| |")
print("| E[n] = E_s[n] - E_H(SIC)[n] |")
print("| = {:.4f} Ha |".format(etot_new))
print("\\------------------------------------/")
break
def Compute_Linear_Covariance(self, comoving):
if self.log.isEnabledFor(logging.INFO):
self.log.info('Computation of prior covariance started')
Covariance=np.zeros((np.size(comoving), np.size(comoving)))
k = np.linspace(10**(-5), 10, 10**6)
size = len(k)//2
Pk = self.Plin(k)
fourier_coeff = np.abs(np.fft.fftn(Pk)[0:size+1])
frqs = np.linspace(0, 0.1*size, size+1)
cf_lin = Spline(frqs, fourier_coeff)
diff=np.zeros((np.size(comoving), np.size(comoving)))
for i in range(np.size(comoving)):
for j in range(np.size(comoving)):
diff[i, j]=np.abs(comoving[i]-comoving[j])
Covariance=cf_lin(diff)
Covariance /= Covariance[0, 0]
if self.log.isEnabledFor(logging.INFO):
self.log.info('Prior Covariance computed')
return Covariance
def update_transfer(self, path_transfer, column=None):
transfer_function = camb2nbodykit(path_transfer, column=column)
transfer_function /= self.max
scales = camb2nbodykit(path_transfer, column=0)
self.transfer_function = Spline(scales, transfer_function)
def __init__(self, cosmo, redshift, transfer, path_transfer=None, column=None):
assert transfer in TRANSFERS
self.transfer = transfer
if self.transfer == 'CAMB':
transfer_function = camb2nbodykit(path_transfer, column=column)
self.max = np.max(transfer_function)
transfer_function /= self.max
scales = camb2nbodykit(path_transfer, column=0)
self.transfer_function = Spline(scales, transfer_function)
# set cosmology values
self._sigma8 = cosmo.sigma8
self.n_s = cosmo.n_s
self.W_T = lambda x: 3/x**3 * (np.sin(x) - x * np.cos(x))
growth = cosmo.scale_independent_growth_factor(redshift)
# normalize to proper sigma8
self._norm = 1
self._norm = (self._sigma8 / self._sigma_r(8.))**2 * growth**2
else:
self.power_spectrum = cosmology.LinearPower(cosmo, redshift, transfer=self.transfer)
def calculate_full_load(full_load_speeds, full_load_powers, idle_engine_speed):
"""
Calculates the full load curve.
:param full_load_speeds:
T1 map speed vector [RPM].
:type full_load_speeds: numpy.array
:param full_load_powers:
T1 map power vector [kW].
:type full_load_powers: numpy.array
:param idle_engine_speed:
Engine speed idle median and std [RPM].
:type idle_engine_speed: (float, float)
:return:
Vehicle full load curve, Maximum power [kW], Rated engine speed [RPM].
:rtype: (scipy.interpolate.InterpolatedUnivariateSpline, float, float)
"""
pn = np.array((full_load_speeds, full_load_powers))
max_speed_at_max_power, max_power = pn[:, np.argmax(pn[1])]
pn[1] /= max_power
idle = idle_engine_speed[0]
pn[0] = (pn[0] - idle) / (max_speed_at_max_power - idle)
return Spline(*pn, ext=3), max_power, max_speed_at_max_power
def redshift_from_distance(cosmo, lg_num_sample=5):
"""Invert redshift-to-distance relationship of a cosmological model
to redshift-from-distance.
Notes
-----
This is useful when the Alcock--Paczynski effect needs to be included
in modelling. Only valid for redshift between 1.e-3 and 100.
Parameters
----------
cosmo : :class:`nbodykit.cosmology.cosmology.Cosmology`
Cosmological model.
lg_num_sample : float, optional
Base-10 logarithm of the number of redshift points to sample
the comoving distance as a function of redshift (default
is 5, i.e. 100000 samle points).
Returns
-------
callable
Redshift-from-distance function.
"""
Z_LOG_RANGE = (-3, 2)
z_samples = np.logspace(*Z_LOG_RANGE, num=10**lg_num_sample)
r_samples = cosmo.comoving_distance(z_samples)
return Spline(r_samples, z_samples, ext='raise')
def spline(x, y, x_eval=None, weights=None, smooth=None):
"""Smoothing spline fit.
The fit and smoothness depend on (i) number of samples, (ii) variance and
(iii) error of each data point.
Parameters
----------
x, y : array-like
The data to fit: y(x).
x_eval : array-like, optional
The points to evaluate the fitted spline. If not given, then the spline
is evaluated on the same original x's.
weights : array-like
Array with 1/std of each data point. Note: to reflect the
heterokedasticity use 1/moving-window-std as weight values.
smooth : float, optional
Smoothing parameter. If weights are given, then s = len(weights).
"""
ind, = np.where((~np.isnan(x)) & (~np.isnan(y)))
x2, y2 = x[ind], y[ind]
if weights is not None:
weights = weights[ind]
if x_eval is None:
x_eval = x
return Spline(x2, y2, w=weights, s=smooth)(x_eval)
def calculate_service_battery_currents_v1(service_battery_capacity, times,
service_battery_state_of_charges):
"""
Calculate the service battery current vector [A].
:param service_battery_capacity:
Service battery capacity [Ah].
:type service_battery_capacity: float
:param times:
Time vector [s].
:type times: numpy.array
:param service_battery_state_of_charges:
State of charge of the service battery [%].
:type service_battery_state_of_charges: numpy.array
:return:
Service battery current vector [A].
:rtype: numpy.array
"""
from scipy.interpolate import UnivariateSpline as Spline
soc = service_battery_state_of_charges
ib = Spline(times, soc, w=np.tile(10, times.shape[0])).derivative()(times)
return ib * (service_battery_capacity * 36.0)
def normalize(energy, veff):
"""integrating with calculated energy eigenvalue and normalization"""
u, r = solve_rad_seq(energy, veff)
u_spline = Spline(r, u * u)
norm = trapezoidal(u_spline, r[0], r[-1])
u2 = u * u / norm
return u2, r, norm
def gaussian_poly_extrap(kout, kint, pint, frac = 1):
'''
Extrapolates beyond the end of kint by a damped polynomial in kint (i.e. Hermite).
Does nothing on the low k end.
The extrapolation form is
(A + B k) * exp(- k^2/k0^2) where k0 is taken to be some fraction (1) of the
final element of kint.
'''
# Solve for the coefficients
k1, k2 = kint[-2], kint[-1]
p1, p2 = pint[-2], pint[-1]
k0 = frac * k2
B = (p2 * np.exp(k2**2/k0**2) - p1 * np.exp(k1**2/k0**2)) / (k2 - k1)
A = p2 * np.exp(k2**2/k0**2) - B * k2
# Interpolate/extrapolate
ret = np.zeros_like(kout)
extrap_iis = (kout > k2)
ret[~extrap_iis] = Spline(kint,pint)(kout[~extrap_iis])
ret[extrap_iis] = ( (A + B*kout) * np.exp(-kout**2/k0**2) )[extrap_iis]
return ret
def interpolate_bb_temps(self):
"""Interpolating the BB H/K temperatures all over the dataframe.
This is necessary, as the frequency of the measurements is too low to have
meaningful mean values during the BB-views. Also, bb_2_temps are measured so
rarely that there might not be at all a measurement during a bb-view.
"""
# just a shortcutting reference
df = self.df
# bb_1_temp is much more often sampled than bb_2_temp
bb1temps = df.bb_1_temp.dropna()
bb2temps = df.bb_2_temp.dropna()
# converting to float because the fitting libraries want to have floats
all_times = df.index.values.astype('float64')
# loop over both temperature arrays [D.R.Y. principle!]
# the number of data points in bb1temps are much higher, but for
# consistency we should interpolate both the same way.
for bbtemp in [bb1temps, bb2temps]:
# converting the time series to floats for interpolation
ind = bbtemp.index.values.astype('float64')
# s=0.0 means I do not allow distance from measured points for the spline
# k=1 means that it will be a local-linear fitted spline between points
# create interpolator function
temp_interpolator = Spline(ind, bbtemp, s=0.0, k=1)
# get new temperatures at all_times
df[bbtemp.name + '_interp'] = temp_interpolator(all_times)
def main():
print("\n/-----------------------------------------------\\")
print("| Ass.5.1: Hartree energy for H-atom GS density |")
print("\\-----------------------------------------------/\n")
print("[INFO] using h = {} for numerically solving ODEs".format(h))
# use secant method for finding the root in u_0(E)
# integration returns (u(r_i), r_i) --> u(r_0) = solver(E)[0][0]
root = newton(lambda en0: solve_rseq(en0)[0][0], -2.0)
print("[INFO] found energy eigenvalue @ E = {:.5f} Ha".format(root))
# integrate again using the correct energy
u, r = solve_rseq(root)
# normalize u**2 to 1
norm = trapz(u ** 2, r)
u /= np.sqrt(norm)
print("[INFO] normalizing |u(r)|^2 from {:.5f} to 1".format(norm))
# interpolate normalized u(r) using B-splines
u_spl = Spline(r, u)
# determine w(r) via Poisson's equation for single-orbital density n_s(r)
w, r = solve_poisson(u_spl)
# find hom.sol. w_hom(r) = a*r to match BC w(r_max) = q_tot = 1 for H-atom
beta = (1 - w[-1]) / r[-1]
w += beta * r
print("[INFO] adding w_hom(r) = b * r --> b = {:.4f}".format(beta))
w_spl = Spline(r, w)
v_spl = Spline(r, w / r)
# compute Hartree energy (verify integration interval choice in plot later)
eh = 0.5 * trapz(w_spl(r) / r * u_spl(r) ** 2, r)
print("[INFO] Hartree energy: {:.5f} Ha".format(eh))
# compare numerical vs. exact Hartree potential energy function
w_exact = lambda r: -(r + 1) * np.exp(-2 * r) + 1 # noqa
v_exact = lambda r: w_exact(r) / r # noqa
plt.plot(r, w_spl(r), lw=2, ls="--", label=r"$w(r)$ num")
plt.plot(r, v_spl(r), lw=2, ls="--", label=r"$v(r)$ num")
plt.plot(r, w_exact(r), lw=3, alpha=0.5, label=r"$w(r)$ exact")
plt.plot(r, v_exact(r), lw=3, alpha=0.5, label=r"$v(r)$ exact")
# integrand from Hartree energy --> integrating up to r=10 is sufficient
label = r"$v(r) \, |u(r)|^2$"
plt.plot(r, v_spl(r) * u_spl(r) ** 2, alpha=0.5, lw=3, label=label)
plt.xlabel("r in Bohr")
plt.legend(loc="best", fancybox=True, shadow=True)
plt.show()
def _interp(r, y, nr):
from scipy.interpolate import InterpolatedUnivariateSpline as Spline
n = len(r)
if n == 0:
return np.nan * nr
elif n == 1:
return (y / r) * nr
else:
return Spline(r, y / r, k=1)(nr) * nr
def smooth_spline(data, key, k, smooth):
import numpy as np
from scipy.interpolate import UnivariateSpline as Spline
span = max(data[key]) - min(data[key])
y = np.array(data[key])
x = np.linspace(0, 1, num=len(y), endpoint=False)
spl = Spline(x, y, k=k, s=smooth * span * k)
return spl
def hotqcd_e3p_T4(T, *args, **kwargs):
"""
Evaluate the trace anomaly (e-3p)/T^4 for the HotQCD EOS.
See Eq. (5) in http://inspirehep.net/record/1307761:
(e-3p)/T^4 = T * d/dT(p/T^4)
"""
# numerically differentiate via interpolating spline
spl = Spline(T, hotqcd_p_T4(T, *args, **kwargs))
return T * spl(T, nu=1)
def interpolate_caldata_worker(self, offset_times, bbcal_times, all_times):
sdata = get_data_columns(self.df)
# create 2 new pd.DataFrames to hold the interpolated gains and offsets
offsets_interp = pd.DataFrame(index=sdata.index)
gains_interp = pd.DataFrame(index=sdata.index)
for det in thermal_detectors:
# change k for the kind of fit you want
s_offset = Spline(offset_times,
self.offsets[det],
s=0.0,
k=self.calfitting_order)
s_gain = Spline(bbcal_times,
self.gains[det],
s=0.0,
k=self.calfitting_order)
offsets_interp[det] = s_offset(all_times)
gains_interp[det] = s_gain(all_times)
return offsets_interp, gains_interp
def get_full_load(ignition_type):
"""
Returns vehicle full load curve.
:param ignition_type:
Engine ignition type (positive or compression).
:type ignition_type: str
:return:
Vehicle normalized full load curve.
:rtype: scipy.interpolate.InterpolatedUnivariateSpline
"""
return Spline(*dfl.functions.get_full_load.FULL_LOAD[ignition_type], ext=3)
def compute_xi_real(self, rr, b1, b2, bs, b3, alpha, alpha_v, alpha_s0,
alpha_s2, s2fog):
'''
Compute the real-space correlation function at rr.
'''
# This is just the zeroth moment:
xieft = self.ximatter + b1*self.xitable[:,1] + b1**2*self.xitable[:,2]\
+ b2*self.xitable[:,3] + b1*b2*self.xitable[:,4]\
+ b2**2 * self.xitable[:,5]\
+ bs*self.xitable[:,6] + b1*bs*self.xitable[:,7]\
+ b2*bs*self.xitable[:,8]\
+ bs**2*self.xitable[:,9] + b3*self.xitable[:,10]\
+ b1*b3*self.xitable[:,11] + alpha*self.xict
xir = Spline(self.rint, xieft)(rr)
return xir