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actionAngleVerticalInverse.py
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actionAngleVerticalInverse.py
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###############################################################################
# actionAngle: a Python module to calculate actions, angles, and frequencies
#
# class: actionAngleVerticalInverse
#
# Calculate (x,v) coordinates for a one-dimensional potential
# given actions-angle coordinates
#
###############################################################################
import copy
import warnings
import numpy
from matplotlib import cm, gridspec, pyplot
from matplotlib.ticker import NullFormatter
from numpy.polynomial import chebyshev, polynomial
from scipy import interpolate, ndimage, optimize
from ..potential import evaluatelinearForces, evaluatelinearPotentials
from ..util import galpyWarning
from ..util import plot as plot
from .actionAngleHarmonic import actionAngleHarmonic
from .actionAngleHarmonicInverse import actionAngleHarmonicInverse
from .actionAngleInverse import actionAngleInverse
from .actionAngleVertical import actionAngleVertical
class actionAngleVerticalInverse(actionAngleInverse):
"""Inverse action-angle formalism for one dimensional systems"""
def __init__(
self,
pot=None,
Es=[0.1, 0.3],
nta=128,
setup_interp=False,
use_pointtransform=False,
pt_deg=7,
pt_nxa=301,
maxiter=100,
angle_tol=1e-12,
bisect=False,
):
"""
Initialize an actionAngleVerticalInverse object
Parameters
----------
pot : Potential object or list of such objects
a linearPotential/verticalPotential or list thereof
Es : numpy.ndarray
energies of the orbits to map the tori for, will be forcibly sorted (needs to be a dense grid when setting up the object for interpolation with setup_interp=True)
nta : int
number of auxiliary angles to sample the torus at when mapping the torus
setup_interp : bool
if True, setup interpolation grids that allow any torus within the E range to be accessed through interpolation
use_pointtransform : bool
if True, use a point transformation to improve the accuracy of the mapping
pt_deg : int
degree of the point transformation polynomial
pt_nxa : int
number of points to use in the point transformation
maxiter : int
maximum number of iterations of root-finding algorithms
angle_tol : float
tolerance for angle root-finding (f(x) is within tol of desired value)
bisect : bool
if True, use simple bisection for root-finding, otherwise first try Newton-Raphson (mainly useful for testing the bisection fallback)
Notes
-----
- 2018-04-11 - Started - Bovy (UofT)
"""
# actionAngleInverse.__init__(self,*args,**kwargs)
if pot is None: # pragma: no cover
raise OSError("Must specify pot= for actionAngleVerticalInverse")
self._pot = pot
self._aAV = actionAngleVertical(pot=self._pot)
# Compute action, frequency, and xmax for each energy
self._nE = len(Es)
js = numpy.empty(self._nE)
Omegas = numpy.empty(self._nE)
xmaxs = numpy.empty(self._nE)
self._Es = numpy.sort(numpy.array(Es))
for ii, E in enumerate(Es):
if (E - evaluatelinearPotentials(self._pot, 0.0)) < 1e-14:
# J=0, should be using vertical freq. from 2nd deriv.
tJ, tO = self._aAV.actionsFreqs(
0.0,
numpy.sqrt(
2.0 * (E + 1e-5 - evaluatelinearPotentials(self._pot, 0.0))
),
)
js[ii] = 0.0
Omegas[ii] = tO[0]
xmaxs[ii] = 0.0
continue
tJ, tO = self._aAV.actionsFreqs(
0.0, numpy.sqrt(2.0 * (E - evaluatelinearPotentials(self._pot, 0.0)))
)
js[ii] = tJ[0]
Omegas[ii] = tO[0]
xmaxs[ii] = self._aAV.calcxmax(
0.0,
numpy.sqrt(2.0 * (E - evaluatelinearPotentials(self._pot, 0.0))),
E=E,
)
self._js = js
self._Omegas = Omegas
self._xmaxs = xmaxs
# Set harmonic-oscillator frequencies == frequencies
self._OmegaHO = copy.copy(Omegas)
# The following work properly for arrays of omega
self._hoaa = actionAngleHarmonic(omega=self._OmegaHO)
self._hoaainv = actionAngleHarmonicInverse(omega=self._OmegaHO)
if use_pointtransform and pt_deg > 1:
self._setup_pointtransform(pt_deg - (1 - pt_deg % 2), pt_nxa) # make odd
else:
# Setup identity point transformation
self._pt_deg = 1
self._pt_nxa = pt_nxa
self._pt_xmaxs = self._xmaxs
self._pt_coeffs = numpy.zeros((self._nE, 2))
self._pt_coeffs[:, 1] = 1.0
self._pt_deriv_coeffs = numpy.ones((self._nE, 1))
self._pt_deriv2_coeffs = numpy.zeros((self._nE, 1))
# Now map all tori
self._nta = nta
self._thetaa = numpy.linspace(0.0, 2.0 * numpy.pi * (1.0 - 1.0 / nta), nta)
self._maxiter = maxiter
self._angle_tol = angle_tol
self._bisect = bisect
self._xgrid = self._create_xgrid()
self._ja = _ja(
self._xgrid,
self._Egrid,
self._pot,
self._omegagrid,
self._ptcoeffsgrid,
self._ptderivcoeffsgrid,
self._xmaxgrid,
self._ptxmaxgrid,
)
self._djadj = (
_djadj(
self._xgrid,
self._Egrid,
self._pot,
self._omegagrid,
self._ptcoeffsgrid,
self._ptderivcoeffsgrid,
self._ptderiv2coeffsgrid,
self._xmaxgrid,
self._ptxmaxgrid,
)
* numpy.atleast_2d(self._Omegas / self._OmegaHO).T
) # In case not 1!
self._djadj[self._js < 1e-10] = 1.0 # J = 0 special case
# Store mean(ja), this is only a better approx. of j w/ no PT!
self._js_orig = copy.copy(self._js)
self._js = numpy.nanmean(self._ja, axis=1)
# Store better approximation to Omega
self._Omegas_orig = copy.copy(self._Omegas)
self._Omegas /= numpy.nanmean(self._djadj, axis=1)
# Compute Fourier expansions
self._nforSn = numpy.arange(self._ja.shape[1] // 2 + 1)
self._nSn = (
numpy.real(
numpy.fft.rfft(
self._ja - numpy.atleast_2d(numpy.nanmean(self._ja, axis=1)).T,
axis=1,
)
)[:, 1:]
/ self._ja.shape[1]
)
self._dSndJ = (
numpy.real(
numpy.fft.rfft(
self._djadj / numpy.atleast_2d(numpy.nanmean(self._djadj, axis=1)).T
- 1.0,
axis=1,
)
)[:, 1:]
/ self._ja.shape[1]
)
# Interpolation of small, noisy coeffs doesn't work, so set to zero
if setup_interp:
self._nSn[numpy.fabs(self._nSn) < 1e-16] = 0.0
self._dSndJ[numpy.fabs(self._dSndJ) < 1e-15] = 0.0
self._dSndJ /= numpy.atleast_2d(self._nforSn)[:, 1:]
self._nforSn = self._nforSn[1:]
self._js[self._Es < 1e-10] = 0.0
# Should use sqrt(2nd deriv. pot), but currently not implemented for 1D
if self._nE > 1:
self._OmegaHO[self._Es < 1e-10] = self._OmegaHO[1]
self._Omegas[self._Es < 1e-10] = self._Omegas[1]
self._nSn[self._js < 1e-10] = 0.0
self._dSndJ[self._js < 1e-10] = 0.0
# Setup interpolation if requested
if setup_interp:
self._interp = True
self._setup_interp()
else:
self._interp = False
return None
def _setup_pointtransform(self, pt_deg, pt_nxa):
# Setup a point transformation for each torus
self._pt_deg = pt_deg
self._pt_nxa = pt_nxa
xamesh = numpy.linspace(-1.0, 1.0, pt_nxa)
self._pt_coeffs = numpy.empty((self._nE, pt_deg + 1))
self._pt_deriv_coeffs = numpy.empty((self._nE, pt_deg))
self._pt_deriv2_coeffs = numpy.empty((self._nE, pt_deg - 1))
self._pt_xmaxs = numpy.sqrt(2.0 * self._js / self._OmegaHO)
for ii in range(self._nE):
if self._js[ii] < 1e-10: # Just use identity for small J
self._pt_coeffs[ii] = 0.0
self._pt_coeffs[ii, 1] = 1.0
self._pt_deriv_coeffs[ii] = 1.0
self._pt_deriv2_coeffs[ii] = 0.0
self._pt_xmaxs[ii] = self._xmaxs[ii] + 1e-10 # avoid /0
coeffs = self._pt_coeffs[ii] # to start next fit
continue
Ea = self._js[ii] * self._OmegaHO[ii]
# Function to optimize with least squares: p-p
def opt_func(coeffs):
# constraints: symmetric, maps [-1,1] --> [-1,1]
ccoeffs = numpy.zeros(pt_deg + 1)
ccoeffs[1] = 1.0
ccoeffs[3::2] = coeffs
ccoeffs /= chebyshev.chebval(1, ccoeffs)
pt = chebyshev.Chebyshev(ccoeffs)
xmesh = pt(xamesh) * self._xmaxs[ii]
# Compute v from (E,xmesh)
v2mesh = 2.0 * (
self._Es[ii] - evaluatelinearPotentials(self._pot, xmesh)
)
v2mesh[v2mesh < 0.0] = 0.0
vmesh = numpy.sqrt(v2mesh)
# Compute v from va = 2(E-HO) and transform
va2mesh = 2.0 * (
Ea
- self._OmegaHO[ii] ** 2.0
* (xamesh * self._pt_xmaxs[ii]) ** 2.0
/ 2.0
)
va2mesh[va2mesh < 0.0] = 0.0
vamesh = numpy.sqrt(va2mesh)
piprime = pt.deriv()(xamesh) * self._xmaxs[ii] / self._pt_xmaxs[ii]
vtildemesh = (
vamesh - numpy.sqrt(v2mesh) * (1.0 / piprime - piprime)
) / piprime
return vmesh - vtildemesh
if ii == 0:
# Start from identity mapping
start_coeffs = [0.0]
start_coeffs.extend([0.0 for jj in range((pt_deg + 1) // 2 - 2)])
else:
# Start from previous best fit
start_coeffs = coeffs[3::2] / coeffs[1]
coeffs = optimize.leastsq(opt_func, start_coeffs)[0]
# Extract full Chebyshev parameters from constrained optimization
ccoeffs = numpy.zeros(pt_deg + 1)
ccoeffs[1] = 1.0
ccoeffs[3::2] = coeffs
ccoeffs /= chebyshev.chebval(1, ccoeffs) # map exact [-1,1] --> [-1,1]
coeffs = ccoeffs
# Store point transformation as simple polynomial
self._pt_coeffs[ii] = chebyshev.cheb2poly(coeffs)
self._pt_deriv_coeffs[ii] = polynomial.polyder(self._pt_coeffs[ii], m=1)
self._pt_deriv2_coeffs[ii] = polynomial.polyder(self._pt_coeffs[ii], m=2)
return None
def _create_xgrid(self):
# Find x grid for regular grid in auxiliary angle (thetaa)
# in practice only need to map 0 < thetaa < pi/2 to +x with +v bc symm
# To efficiently start the search, we first compute thetaa for a dense
# grid in x (at +v)
xgrid = numpy.linspace(-1.0, 1.0, 2 * self._nta)
xs = xgrid * numpy.atleast_2d(self._pt_xmaxs).T
xta = _anglea(
xs,
numpy.tile(self._Es, (xs.shape[1], 1)).T,
self._pot,
numpy.tile(self._hoaa._omega, (xs.shape[1], 1)).T,
numpy.rollaxis(numpy.tile(self._pt_coeffs, (xs.shape[1], 1, 1)), 1),
numpy.rollaxis(numpy.tile(self._pt_deriv_coeffs, (xs.shape[1], 1, 1)), 1),
numpy.tile(self._xmaxs, (xs.shape[1], 1)).T,
numpy.tile(self._pt_xmaxs, (xs.shape[1], 1)).T,
)
xta[numpy.isnan(xta)] = 0.0 # Zero energy orbit -> NaN
# Now use Newton-Raphson to iterate to a regular grid
cindx = numpy.nanargmin(
numpy.fabs(
(xta - numpy.rollaxis(numpy.atleast_3d(self._thetaa), 1) + numpy.pi)
% (2.0 * numpy.pi)
- numpy.pi
),
axis=2,
)
xgrid = xgrid[cindx].T * numpy.atleast_2d(self._pt_xmaxs).T
Egrid = numpy.tile(self._Es, (self._nta, 1)).T
omegagrid = numpy.tile(self._hoaa._omega, (self._nta, 1)).T
xmaxgrid = numpy.tile(self._xmaxs, (self._nta, 1)).T
ptxmaxgrid = numpy.tile(self._pt_xmaxs, (self._nta, 1)).T
ptcoeffsgrid = numpy.rollaxis(numpy.tile(self._pt_coeffs, (self._nta, 1, 1)), 1)
ptderivcoeffsgrid = numpy.rollaxis(
numpy.tile(self._pt_deriv_coeffs, (self._nta, 1, 1)), 1
)
ptderiv2coeffsgrid = numpy.rollaxis(
numpy.tile(self._pt_deriv2_coeffs, (self._nta, 1, 1)), 1
)
ta = _anglea(
xgrid,
Egrid,
self._pot,
omegagrid,
ptcoeffsgrid,
ptderivcoeffsgrid,
xmaxgrid,
ptxmaxgrid,
)
mta = numpy.tile(self._thetaa, (len(self._Es), 1))
# Now iterate
cntr = 0
unconv = numpy.ones(xgrid.shape, dtype="bool")
# We'll fill in the -v part using the +v, also remove the endpoints
unconv[:, self._nta // 4 : 3 * self._nta // 4 + 1] = False
# Also don't bother with J=0 torus
unconv[numpy.tile(self._js, (self._nta, 1)).T < 1e-10] = False
dta = (ta[unconv] - mta[unconv] + numpy.pi) % (2.0 * numpy.pi) - numpy.pi
unconv[unconv] = numpy.fabs(dta) > self._angle_tol
# Don't allow too big steps
maxdx = numpy.tile(self._pt_xmaxs / float(self._nta), (self._nta, 1)).T
while not self._bisect:
dtadx = _danglea(
xgrid[unconv],
Egrid[unconv],
self._pot,
omegagrid[unconv],
ptcoeffsgrid[unconv],
ptderivcoeffsgrid[unconv],
ptderiv2coeffsgrid[unconv],
xmaxgrid[unconv],
ptxmaxgrid[unconv],
)
dta = (ta[unconv] - mta[unconv] + numpy.pi) % (2.0 * numpy.pi) - numpy.pi
dx = -dta / dtadx
dx[numpy.fabs(dx) > maxdx[unconv]] = (numpy.sign(dx) * maxdx[unconv])[
numpy.fabs(dx) > maxdx[unconv]
]
xgrid[unconv] += dx
xgrid[unconv * (xgrid > ptxmaxgrid)] = ptxmaxgrid[
unconv * (xgrid > ptxmaxgrid)
]
xgrid[unconv * (xgrid < -ptxmaxgrid)] = ptxmaxgrid[
unconv * (xgrid < -ptxmaxgrid)
]
newta = _anglea(
xgrid[unconv],
Egrid[unconv],
self._pot,
omegagrid[unconv],
ptcoeffsgrid[unconv],
ptderivcoeffsgrid[unconv],
xmaxgrid[unconv],
ptxmaxgrid[unconv],
)
ta[unconv] = newta
unconv[unconv] = numpy.fabs(dta) > self._angle_tol
cntr += 1
if numpy.sum(unconv) == 0:
break
if cntr > self._maxiter:
warnings.warn(
"Torus mapping with Newton-Raphson did not converge in {} iterations, falling back onto simple bisection (increase maxiter to try harder with Newton-Raphson)".format(
self._maxiter
),
galpyWarning,
)
break
if self._bisect or cntr > self._maxiter:
# Reset cntr
cntr = 0
# Start from nearest guess from below
new_xgrid = numpy.linspace(-1.0, 1.0, 2 * self._nta)
da = (
xta - numpy.rollaxis(numpy.atleast_3d(self._thetaa), 1) + numpy.pi
) % (2.0 * numpy.pi) - numpy.pi
da[da >= 0.0] = -numpy.nanmax(numpy.fabs(da)) - 0.1
cindx = numpy.nanargmax(da, axis=2)
tryx_min = (new_xgrid[cindx].T * numpy.atleast_2d(self._pt_xmaxs).T)[unconv]
dx = (
2.0 / (2.0 * self._nta - 1) * ptxmaxgrid
) # delta of initial x grid above
while True:
dx *= 0.5
xgrid[unconv] = tryx_min + dx[unconv]
newta = (
_anglea(
xgrid[unconv],
Egrid[unconv],
self._pot,
omegagrid[unconv],
ptcoeffsgrid[unconv],
ptderivcoeffsgrid[unconv],
xmaxgrid[unconv],
ptxmaxgrid[unconv],
)
+ 2.0 * numpy.pi
) % (2.0 * numpy.pi)
ta[unconv] = newta
dta = (newta - mta[unconv] + numpy.pi) % (2.0 * numpy.pi) - numpy.pi
tryx_min[newta < mta[unconv]] = xgrid[unconv][newta < mta[unconv]]
unconv[unconv] = numpy.fabs(dta) > self._angle_tol
tryx_min = tryx_min[numpy.fabs(dta) > self._angle_tol]
cntr += 1
if numpy.sum(unconv) == 0:
break
if cntr > self._maxiter:
warnings.warn(
"Torus mapping with bisection did not converge in {} iterations".format(
self._maxiter
)
+ " for energies:"
+ "".join(f" {k:g}" for k in sorted(set(Egrid[unconv]))),
galpyWarning,
)
break
xgrid[:, self._nta // 4 + 1 : self._nta // 2 + 1] = xgrid[:, : self._nta // 4][
:, ::-1
]
xgrid[:, self._nta // 2 + 1 : 3 * self._nta // 4 + 1] = xgrid[
:, 3 * self._nta // 4 :
][:, ::-1]
ta[:, self._nta // 4 + 1 : 3 * self._nta // 4] = _anglea(
xgrid[:, self._nta // 4 + 1 : 3 * self._nta // 4],
Egrid[:, self._nta // 4 + 1 : 3 * self._nta // 4],
self._pot,
omegagrid[:, self._nta // 4 + 1 : 3 * self._nta // 4],
ptcoeffsgrid[:, self._nta // 4 + 1 : 3 * self._nta // 4],
ptderivcoeffsgrid[:, self._nta // 4 + 1 : 3 * self._nta // 4],
xmaxgrid[:, self._nta // 4 + 1 : 3 * self._nta // 4],
ptxmaxgrid[:, self._nta // 4 + 1 : 3 * self._nta // 4],
vsign=-1.0,
)
self._dta = (ta - mta + numpy.pi) % (2.0 * numpy.pi) - numpy.pi
self._mta = mta
# Store these, they are useful (obv. arbitrary to return xgrid
# and not just store it...)
self._Egrid = Egrid
self._omegagrid = omegagrid
self._ptcoeffsgrid = ptcoeffsgrid
self._ptderivcoeffsgrid = ptderivcoeffsgrid
self._ptderiv2coeffsgrid = ptderiv2coeffsgrid
self._ptxmaxgrid = ptxmaxgrid
self._xmaxgrid = xmaxgrid
return xgrid
def plot_convergence(
self, E, overplot=False, return_gridspec=False, shift_action=None
):
if shift_action is None:
shift_action = self._pt_deg > 1
# First find the torus for this energy
indx = numpy.nanargmin(numpy.fabs(E - self._Es))
if numpy.fabs(E - self._Es[indx]) > 1e-10:
raise ValueError(
"Given energy not found; please specify an energy used in the initialization of the instance"
)
if not overplot:
gs = gridspec.GridSpec(2, 3, height_ratios=[4, 1])
else:
gs = overplot # confusingly, we overload the meaning of overplot
# mapping of thetaa --> x
pyplot.subplot(gs[0])
plot.plot(
self._thetaa,
self._xgrid[indx],
color="k",
ls="--" if overplot else "-",
ylabel=r"$x(\theta^A)$",
gcf=True,
overplot=overplot,
)
if not overplot:
pyplot.gca().xaxis.set_major_formatter(NullFormatter())
if not overplot:
pyplot.subplot(gs[3])
negv = (self._thetaa > numpy.pi / 2.0) * (
self._thetaa < 3.0 * numpy.pi / 2.0
)
thetaa_out = numpy.empty_like(self._thetaa)
one = numpy.ones(numpy.sum(True ^ negv))
thetaa_out[True ^ negv] = _anglea(
self._xgrid[indx][True ^ negv],
E,
self._pot,
self._OmegaHO[indx],
self._pt_coeffs[indx],
numpy.tile(self._pt_deriv_coeffs[indx], (numpy.sum(True ^ negv), 1)),
self._xmaxs[indx] * one,
self._pt_xmaxs[indx] * one,
vsign=1.0,
)
one = numpy.ones(numpy.sum(negv))
thetaa_out[negv] = _anglea(
self._xgrid[indx][negv],
E,
self._pot,
self._OmegaHO[indx],
self._pt_coeffs[indx],
numpy.tile(self._pt_deriv_coeffs[indx], (numpy.sum(negv), 1)),
self._xmaxs[indx] * one,
self._pt_xmaxs[indx] * one,
vsign=-1.0,
)
plot.plot(
self._thetaa,
((thetaa_out - self._thetaa + numpy.pi) % (2.0 * numpy.pi)) - numpy.pi,
color="k",
gcf=True,
xlabel=r"$\theta^A$",
ylabel=r"$\theta^A[x(\theta^A)]-\theta^A$",
)
# Recovery of the nSn from J^A(theta^A) behavior
pyplot.subplot(gs[1])
plot.plot(
self._thetaa,
self._ja[indx],
color="k",
ls="--" if overplot else "-",
ylabel=r"$J^A(\theta^A),J$",
gcf=True,
overplot=overplot,
)
pyplot.axhline(
self._js[indx] + shift_action * (self._js_orig[indx] - self._js[indx]),
color="k",
ls="--",
)
if not overplot:
pyplot.gca().xaxis.set_major_formatter(NullFormatter())
if not overplot:
pyplot.subplot(gs[4])
plot.plot(
self._thetaa,
numpy.array(
[
self._js[indx]
+ 2.0 * numpy.sum(self._nSn[indx] * numpy.cos(self._nforSn * x))
for x in self._thetaa
]
)
/ self._ja[indx]
- 1.0,
color="k",
xlabel=r"$\theta^A$",
ylabel=r"$\delta J^A/J^A$",
gcf=True,
)
# Recovery of the dSndJ from dJ^A/dJ(theta^A) behavior
pyplot.subplot(gs[2])
plot.plot(
self._thetaa,
self._djadj[indx] / numpy.nanmean(self._djadj[indx]),
color="k",
ls="--" if overplot else "-",
ylabel=r"$\mathrm{d}J^A/\mathrm{d}J(\theta^A)$",
gcf=True,
overplot=overplot,
)
pyplot.axhline(1.0, color="k", ls="--")
if not overplot:
pyplot.gca().xaxis.set_major_formatter(NullFormatter())
if not overplot:
pyplot.subplot(gs[5])
plot.plot(
self._thetaa,
numpy.array(
[
1.0
+ 2.0
* numpy.sum(
self._nforSn
* self._dSndJ[indx]
* numpy.cos(self._nforSn * x)
)
for x in self._thetaa
]
)
- self._djadj[indx] / numpy.nanmean(self._djadj[indx]),
color="k",
xlabel=r"$\theta^A$",
ylabel=r"$\delta \mathrm{d}J^A/\mathrm{d}J(\theta^A)$",
gcf=True,
)
pyplot.tight_layout()
if return_gridspec:
return gs
else:
return None
def plot_power(self, Es, symm=True, overplot=False, return_gridspec=False, ls="-"):
Es = numpy.sort(numpy.atleast_1d(Es))
minn_for_cmap = 4
if len(Es) < minn_for_cmap:
if not overplot:
gs = gridspec.GridSpec(1, 2)
else:
gs = overplot # confusingly, we overload the meaning of overplot
else:
if not overplot:
outer = gridspec.GridSpec(1, 2, width_ratios=[2.0, 0.05], wspace=0.05)
gs = gridspec.GridSpecFromSubplotSpec(
1, 2, subplot_spec=outer[0], wspace=0.35
)
else:
raise RuntimeError(
f"plot_power with >= {minn_for_cmap} energies and overplot=True is not supported"
)
for ii, E in enumerate(Es):
# First find the torus for this energy
indx = numpy.nanargmin(numpy.fabs(E - self._Es))
if numpy.fabs(E - self._Es[indx]) > 1e-10:
raise ValueError(
"Given energy not found; please specify an energy used in the initialization of the instance"
)
# n S_n
y = numpy.fabs(self._nSn[indx, symm :: symm + 1])
if len(Es) > 1 and E == Es[0]:
y4minmax = numpy.fabs(self._nSn[:, symm :: symm + 1])
ymin = numpy.amax(
[numpy.amin(y4minmax[numpy.isfinite(y4minmax)]), 1e-17]
)
ymax = numpy.amax(y4minmax[numpy.isfinite(y4minmax)])
elif len(Es) == 1:
ymin = numpy.amax([numpy.amin(y[numpy.isfinite(y)]), 1e-17])
ymax = numpy.amax(y[numpy.isfinite(y)])
if len(Es) < minn_for_cmap:
label = rf"$E = {E:g}$"
color = f"C{ii}"
else:
label = None
color = cm.plasma((E - Es[0]) / (Es[-1] - Es[0]))
pyplot.subplot(gs[0])
plot.plot(
numpy.fabs(self._nforSn[symm :: symm + 1]),
y,
yrange=[ymin, ymax],
ls=ls,
gcf=True,
semilogy=True,
overplot=overplot,
xrange=[0.0, self._nforSn[-1]],
label=label,
color=color,
xlabel=r"$n$",
ylabel=r"$|nS_n|$",
)
# d S_n / d J
y = numpy.fabs(self._dSndJ[indx, symm :: symm + 1])
if len(Es) > 1 and E == Es[0]:
y4minmax = numpy.fabs(self._dSndJ[:, symm :: symm + 1])
ymin = numpy.amax(
[numpy.amin(y4minmax[numpy.isfinite(y4minmax)]), 1e-17]
)
ymax = numpy.amax(y4minmax[numpy.isfinite(y4minmax)])
elif len(Es) == 1:
ymin = numpy.amax([numpy.amin(y[numpy.isfinite(y)]), 1e-17])
ymax = numpy.amax(y[numpy.isfinite(y)])
if len(Es) < minn_for_cmap:
label = rf"$E = {E:g}$"
color = f"C{ii}"
else:
label = None
color = cm.plasma((E - Es[0]) / (Es[-1] - Es[0]))
pyplot.subplot(gs[1])
plot.plot(
numpy.fabs(self._nforSn[symm :: symm + 1]),
y,
yrange=[ymin, ymax],
ls=ls,
gcf=True,
semilogy=True,
overplot=overplot,
xrange=[0.0, self._nforSn[-1]],
label=label,
color=color,
xlabel=r"$n$",
ylabel=r"$|\mathrm{d}S_n/\mathrm{d}J|$",
)
if not overplot == gs:
overplot = True
if len(Es) < minn_for_cmap:
if not overplot == gs:
pyplot.subplot(gs[0])
pyplot.legend(fontsize=17.0, frameon=False)
pyplot.subplot(gs[1])
pyplot.legend(fontsize=17.0, frameon=False)
pyplot.tight_layout()
else:
pyplot.subplot(outer[1])
sm = pyplot.cm.ScalarMappable(
cmap=cm.plasma, norm=pyplot.Normalize(vmin=Es[0], vmax=Es[-1])
)
sm._A = []
cbar = pyplot.colorbar(
sm, cax=pyplot.gca(), use_gridspec=True, format=r"$%g$"
)
cbar.set_label(r"$E$")
outer.tight_layout(pyplot.gcf())
if return_gridspec:
return gs
else:
return None
def plot_orbit(self, E):
ta = numpy.linspace(0.0, 2.0 * numpy.pi, 1001)
if not self._interp:
# First find the torus for this energy
indx = numpy.nanargmin(numpy.fabs(E - self._Es))
if numpy.fabs(E - self._Es[indx]) > 1e-10:
raise ValueError(
"Given energy not found; please specify an energy used in the initialization of the instance"
)
tJ = self._js[indx]
else:
tJ = self.J(E)
x, v = self(tJ, ta)
# First plot orbit in x,v
pyplot.subplot(1, 2, 1)
plot.plot(
x,
v,
xlabel=r"$x$",
ylabel=r"$v$",
gcf=True,
color="k",
xrange=[1.1 * numpy.amin(x), 1.1 * numpy.amax(x)],
yrange=[1.1 * numpy.amin(v), 1.1 * numpy.amax(v)],
)
# Then plot energy
pyplot.subplot(1, 2, 2)
Eorbit = (v**2.0 / 2.0 + evaluatelinearPotentials(self._pot, x)) / E - 1.0
ymin, ymax = numpy.amin(Eorbit), numpy.amax(Eorbit)
plot.plot(
ta,
Eorbit,
xrange=[0.0, 2.0 * numpy.pi],
yrange=[ymin - (ymax - ymin) * 3.0, ymax + (ymax - ymin) * 3.0],
gcf=True,
color="k",
xlabel=r"$\theta$",
ylabel=r"$E/E_{\mathrm{true}}-1$",
)
pyplot.tight_layout()
return None
################### FUNCTIONS FOR INTERPOLATION BETWEEN TORI###############
def _setup_interp(self):
self._Emin = self._Es[0]
self._Emax = self._Es[-1]
self._nnSn = self._nSn.shape[1] # won't be confusing...
self._nSnNormalize = numpy.ones(self._nnSn)
self._nSnFiltered = ndimage.spline_filter(self._nSn, order=3)
self._dSndJFiltered = ndimage.spline_filter(self._dSndJ, order=3)
self.J = interpolate.InterpolatedUnivariateSpline(self._Es, self._js, k=3)
self.E = interpolate.InterpolatedUnivariateSpline(self._js, self._Es, k=3)
self.OmegaHO = interpolate.InterpolatedUnivariateSpline(
self._Es, self._OmegaHO, k=3
)
self.Omega = interpolate.InterpolatedUnivariateSpline(
self._Es, self._Omegas, k=3
)
self.xmax = interpolate.InterpolatedUnivariateSpline(self._Es, self._xmaxs, k=3)
self.ptxmax = interpolate.InterpolatedUnivariateSpline(
self._Es, self._pt_xmaxs, k=3
)
self._nptcoeffs = self._pt_coeffs.shape[1]
self._ptcoeffsFiltered = ndimage.spline_filter(self._pt_coeffs, order=3)
self._ptderivcoeffsFiltered = ndimage.spline_filter(
self._pt_deriv_coeffs, order=3
)
return None
def _coords_for_map_coords(self, E):
coords = numpy.empty((2, self._nnSn * len(E)))
coords[0] = numpy.tile(
(E - self._Emin) / (self._Emax - self._Emin) * (self._nE - 1.0),
(self._nnSn, 1),
).T.flatten()
coords[1] = numpy.tile(self._nforSn - 1, (len(E), 1)).flatten()
return coords
def nSn(self, E):
if not self._interp:
raise RuntimeError(
"To evaluate nSn, interpolation must be activated at instantiation using setup_interp=True"
)
evalE = numpy.atleast_1d(E)
indxc = (evalE >= self._Emin) * (evalE <= self._Emax)
coords = self._coords_for_map_coords(evalE[indxc])
out = numpy.empty((len(evalE), self._nnSn))
out[indxc] = numpy.reshape(
ndimage.map_coordinates(
self._nSnFiltered, coords, order=3, prefilter=False
),
(numpy.sum(indxc), self._nnSn),
)
out[True ^ indxc] = numpy.nan
return out
def dSndJ(self, E):
if not self._interp:
raise RuntimeError(
"To evaluate dnSndJ, interpolation must be activated at instantiation using setup_interp=True"
)
evalE = numpy.atleast_1d(E)
indxc = (evalE >= self._Emin) * (evalE <= self._Emax)
coords = self._coords_for_map_coords(evalE[indxc])
out = numpy.empty((len(evalE), self._nnSn))
out[indxc] = numpy.reshape(
ndimage.map_coordinates(
self._dSndJFiltered, coords, order=3, prefilter=False
),
(numpy.sum(indxc), self._nnSn),
)
out[True ^ indxc] = numpy.nan
return out
def _coords_for_map_coords_pt(self, E, deriv=False):
coords = numpy.empty((2, (self._nptcoeffs - deriv) * len(E)))
coords[0] = numpy.tile(
(E - self._Emin) / (self._Emax - self._Emin) * (self._nE - 1.0),
(self._nptcoeffs - deriv, 1),
).T.flatten()
coords[1] = numpy.tile(
numpy.arange(self._nptcoeffs - deriv), (len(E), 1)
).flatten()
return coords
def pt_coeffs(self, E):
if not self._interp:
raise RuntimeError(
"To evaluate pt_coeffs, interpolation must be activated at instantiation using setup_interp=True"
)
evalE = numpy.atleast_1d(E)
indxc = (evalE >= self._Emin) * (evalE <= self._Emax)
coords = self._coords_for_map_coords_pt(evalE[indxc], deriv=False)
out = numpy.empty((len(evalE), self._nptcoeffs))
out[indxc] = numpy.reshape(
ndimage.map_coordinates(
self._ptcoeffsFiltered, coords, order=3, prefilter=False
),
(numpy.sum(indxc), self._nptcoeffs),
)
out[True ^ indxc] = numpy.nan
return out
def pt_deriv_coeffs(self, E):
if not self._interp:
raise RuntimeError(
"To evaluate pt_deriv_coeffs, interpolation must be activated at instantiation using setup_interp=True"
)
evalE = numpy.atleast_1d(E)
indxc = (evalE >= self._Emin) * (evalE <= self._Emax)
coords = self._coords_for_map_coords_pt(evalE[indxc], deriv=True)
out = numpy.empty((len(evalE), self._nptcoeffs - 1))
out[indxc] = numpy.reshape(
ndimage.map_coordinates(
self._ptderivcoeffsFiltered, coords, order=3, prefilter=False
),
(numpy.sum(indxc), self._nptcoeffs - 1),
)
out[True ^ indxc] = numpy.nan
return out
def plot_interp(self, E, symm=True):
truthaAV = actionAngleVerticalInverse(
pot=self._pot,
Es=[E],
nta=self._nta,
setup_interp=False,
use_pointtransform=self._pt_deg > 1,
pt_deg=self._pt_deg,
pt_nxa=self._pt_nxa,
)
# Check whether S_n is matched
pyplot.subplot(2, 3, 1)
y = numpy.fabs(self.nSn(E)[0, symm :: symm + 1])
ymin = numpy.amax([numpy.amin(y[numpy.isfinite(y)]), 1e-17])
ymax = numpy.amax(y[numpy.isfinite(y)])
plot.plot(
numpy.fabs(self._nforSn[symm :: symm + 1]),
y,
yrange=[ymin, ymax],
gcf=True,
semilogy=True,
xrange=[0.0, self._nforSn[-1]],
label=r"$\mathrm{Interpolation}$",
xlabel=r"$n$",
ylabel=r"$|nS_n|$",
)
plot.plot(
self._nforSn[symm :: symm + 1],
truthaAV._nSn[0, symm :: symm + 1],
overplot=True,
label=r"$\mathrm{Direct}$",
)
pyplot.legend(fontsize=17.0, frameon=False)
pyplot.subplot(2, 3, 4)
y = ((self.nSn(E)[0] - truthaAV._nSn[0]) / truthaAV._nSn[0])[symm :: symm + 1]
ymin = numpy.amin(y[numpy.isfinite(y)])
ymax = numpy.amax(y[numpy.isfinite(y)])
plot.plot(
self._nforSn[symm :: symm + 1],
y,
yrange=[ymin, ymax],
xrange=[0.0, self._nforSn[-1]],
gcf=True,
xlabel=r"$n$",
ylabel=r"$S_{n,\mathrm{interp}}/S_{n,\mathrm{direct}}-1$",
)
# Check whether d S_n / d J is matched
pyplot.subplot(2, 3, 2)
y = numpy.fabs(self.dSndJ(E)[0, symm :: symm + 1])
ymin = numpy.amax([numpy.amin(y[numpy.isfinite(y)]), 1e-18])
ymax = numpy.amax(y[numpy.isfinite(y)])
plot.plot(
numpy.fabs(self._nforSn[symm :: symm + 1]),
y,
yrange=[ymin, ymax],
xrange=[0.0, self._nforSn[-1]],
gcf=True,
semilogy=True,
label=r"$\mathrm{Interpolation}$",
xlabel=r"$n$",
ylabel=r"$|\mathrm{d}S_n/\mathrm{d}J|$",
)
plot.plot(
self._nforSn[symm :: symm + 1],
numpy.fabs(truthaAV._dSndJ[0, symm :: symm + 1]),
overplot=True,
label=r"$\mathrm{Direct}$",
)
pyplot.legend(fontsize=17.0, frameon=False)
pyplot.subplot(2, 3, 5)
y = ((self.dSndJ(E)[0] - truthaAV._dSndJ[0]) / truthaAV._dSndJ[0])[
symm :: symm + 1
]
ymin = numpy.amin(y[numpy.isfinite(y)])
ymax = numpy.amax(y[numpy.isfinite(y)])
plot.plot(
self._nforSn[symm :: symm + 1],
y,
yrange=[ymin, ymax],
xrange=[0.0, self._nforSn[-1]],
gcf=True,
xlabel=r"$n$",
ylabel=r"$(\mathrm{d}S_n/\mathrm{d}J)_{\mathrm{interp}}/(\mathrm{d}S_n/\mathrm{d}J)_{\mathrm{direct}}-1$",
)
# Check energy along the torus
pyplot.subplot(2, 3, 3)
ta = numpy.linspace(0.0, 2.0 * numpy.pi, 1001)
x, v = truthaAV(truthaAV._js, ta)
Edirect = v**2.0 / 2.0 + evaluatelinearPotentials(self._pot, x)
x, v = self(self.J(E), ta)
Einterp = v**2.0 / 2.0 + evaluatelinearPotentials(self._pot, x)
ymin, ymax = numpy.amin([Edirect, Einterp]), numpy.amax([Edirect, Einterp])
plot.plot(
ta,
Einterp,
xrange=[0.0, 2.0 * numpy.pi],
yrange=[ymin - (ymax - ymin) * 2.0, ymax + (ymax - ymin) * 2.0],
gcf=True,
label=r"$\mathrm{Interpolation}$",
xlabel=r"$\theta$",
ylabel=r"$E$",
)
plot.plot(ta, Edirect, overplot=True, label=r"$\mathrm{Direct}$")
pyplot.legend(fontsize=17.0, frameon=False)
pyplot.subplot(2, 3, 6)
plot.plot(
ta,
Einterp / Edirect - 1.0,
xrange=[0.0, 2.0 * numpy.pi],
gcf=True,
label=r"$\mathrm{Interpolation}$",
xlabel=r"$\theta$",
ylabel=r"$E_{\mathrm{interp}}/E_{\mathrm{direct}}-1$",
)
pyplot.tight_layout()
return None
def J(self, E):