class LensModel(object):
"""
class to handle an arbitrary list of lens models
"""
def __init__(self,
lens_model_list,
z_lens=None,
z_source=None,
lens_redshift_list=None,
cosmo=None,
multi_plane=False,
numerical_alpha_class=None,
observed_convention_index=None,
z_source_convention=None):
"""
:param lens_model_list: list of strings with lens model names
:param z_lens: redshift of the deflector (only considered when operating in single plane mode).
Is only needed for specific functions that require a cosmology.
:param z_source: redshift of the source: Needed in multi_plane option only,
not required for the core functionalities in the single plane mode.
:param lens_redshift_list: list of deflector redshift (corresponding to the lens model list),
only applicable in multi_plane mode.
:param cosmo: instance of the astropy cosmology class. If not specified, uses the default cosmology.
:param multi_plane: bool, if True, uses multi-plane mode. Default is False.
:param numerical_alpha_class: an instance of a custom class for use in NumericalAlpha() lens model
(see documentation in Profiles/numerical_alpha)
:param observed_convention_index: a list of lens indexes that correspond to observed positions on the sky, not
physical positions
:param z_source_convention: float, redshift of a source to define the reduced deflection angles of the lens
models. If None, 'z_source' is used.
"""
self.lens_model_list = lens_model_list
self.z_lens = z_lens
if z_source_convention is None:
z_source_convention = z_source
self.z_source = z_source
self._z_source_convention = z_source_convention
self.redshift_list = lens_redshift_list
if cosmo is None:
cosmo = default_cosmology.get()
self.cosmo = cosmo
self.multi_plane = multi_plane
if multi_plane is True:
if z_source is None:
raise ValueError(
'z_source needs to be set for multi-plane lens modelling.')
self.lens_model = MultiPlane(
z_source,
lens_model_list,
lens_redshift_list,
cosmo=cosmo,
numerical_alpha_class=numerical_alpha_class,
observed_convention_index=observed_convention_index,
z_source_convention=z_source_convention)
else:
self.lens_model = SinglePlane(
lens_model_list,
numerical_alpha_class=numerical_alpha_class,
lens_redshift_list=lens_redshift_list,
z_source_convention=z_source_convention)
if z_lens is not None and z_source is not None:
self._lensCosmo = LensCosmo(z_lens, z_source, cosmo=cosmo)
def ray_shooting(self, x, y, kwargs, k=None):
"""
maps image to source position (inverse deflection)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: source plane positions corresponding to (x, y) in the image plane
"""
return self.lens_model.ray_shooting(x, y, kwargs, k=k)
def fermat_potential(self, x_image, y_image, x_source, y_source,
kwargs_lens):
"""
fermat potential (negative sign means earlier arrival time)
:param x_image: image position
:param y_image: image position
:param x_source: source position
:param y_source: source position
:param kwargs_lens: list of keyword arguments of lens model parameters matching the lens model classes
:return: fermat potential in arcsec**2 without geometry term (second part of Eqn 1 in Suyu et al. 2013) as a list
"""
if hasattr(self.lens_model, 'fermat_potential'):
return self.lens_model.fermat_potential(x_image, y_image, x_source,
y_source, kwargs_lens)
else:
raise ValueError(
"Fermat potential is not defined in multi-plane lensing. Please use single plane lens models."
)
def arrival_time(self, x_image, y_image, kwargs_lens):
"""
:param x_image: image position
:param y_image: image position
:param kwargs_lens: lens model parameter keyword argument list
:return: arrival time of image positions in units of days
"""
try:
arrival_time = self.lens_model.arrival_time(
x_image, y_image, kwargs_lens)
except:
x_source, y_source = self.lens_model.ray_shooting(
x_image, y_image, kwargs_lens)
fermat_pot = self.lens_model.fermat_potential(
x_image, y_image, x_source, y_source, kwargs_lens)
if not hasattr(self, '_lensCosmo'):
raise ValueError(
"LensModel class was not initalized with lens and source redshifts!"
)
arrival_time = self._lensCosmo.time_delay_units(fermat_pot)
return arrival_time
def potential(self, x, y, kwargs, k=None):
"""
lensing potential
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: lensing potential in units of arcsec^2
"""
return self.lens_model.potential(x, y, kwargs, k=k)
def alpha(self, x, y, kwargs, k=None):
"""
deflection angles
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: deflection angles in units of arcsec
"""
return self.lens_model.alpha(x, y, kwargs, k=k)
def hessian(self, x, y, kwargs, k=None):
"""
hessian matrix
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: f_xx, f_xy, f_yy components
"""
return self.lens_model.hessian(x, y, kwargs, k=k)
def kappa(self, x, y, kwargs, k=None):
"""
lensing convergence k = 1/2 laplacian(phi)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: lensing convergence
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x, y, kwargs, k=k)
kappa = 1. / 2 * (f_xx + f_yy)
return kappa
def gamma(self, x, y, kwargs, k=None):
"""
shear computation
g1 = 1/2(d^2phi/dx^2 - d^2phi/dy^2)
g2 = d^2phi/dxdy
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: gamma1, gamma2
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x, y, kwargs, k=k)
gamma1 = 1. / 2 * (f_xx - f_yy)
gamma2 = f_xy
return gamma1, gamma2
def magnification(self, x, y, kwargs, k=None):
"""
magnification
mag = 1/det(A)
A = 1 - d^2phi/d_ij
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: magnification
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x, y, kwargs, k=k)
det_A = (1 - f_xx) * (1 - f_yy) - f_xy * f_yx
return 1. / det_A # attention, if dividing by zero
def flexion(self, x, y, kwargs, diff=0.000001):
"""
third derivatives (flexion)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param diff: numerical differential length of Hessian
:return: f_xxx, f_xxy, f_xyy, f_yyy
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x, y, kwargs)
f_xx_dx, f_xy_dx, f_yx_dx, f_yy_dx = self.hessian(x + diff, y, kwargs)
f_xx_dy, f_xy_dy, f_yx_dy, f_yy_dy = self.hessian(x, y + diff, kwargs)
f_xxx = (f_xx_dx - f_xx) / diff
f_xxy = (f_xx_dy - f_xx) / diff
f_xyy = (f_xy_dy - f_xy) / diff
f_yyy = (f_yy_dy - f_yy) / diff
return f_xxx, f_xxy, f_xyy, f_yyy
def set_static(self, kwargs):
"""
set this instance to a static lens model. This can improve the speed in evaluating lensing quantities at
different positions but must not be used with different lens model parameters!
:param kwargs: lens model keyword argument list
:return: kwargs_updated (in case of image position convention in multiplane lensing this is changed)
"""
return self.lens_model.set_static(kwargs)
def set_dynamic(self):
"""
deletes cache for static setting and makes sure the observed convention in the position of lensing profiles in
the multiplane setting is enabled. Dynamic is the default setting of this class enabling an accurate computation
of lensing quantities with different parameters in the lensing profiles.
:return: None
"""
self.lens_model.set_dynamic()
class LensModel(object):
"""
class to handle an arbitrary list of lens models. This is the main lenstronomy LensModel API for all other modules.
"""
def __init__(self,
lens_model_list,
z_lens=None,
z_source=None,
lens_redshift_list=None,
cosmo=None,
multi_plane=False,
numerical_alpha_class=None,
observed_convention_index=None,
z_source_convention=None,
cosmo_interp=False,
z_interp_stop=None,
num_z_interp=100):
"""
:param lens_model_list: list of strings with lens model names
:param z_lens: redshift of the deflector (only considered when operating in single plane mode).
Is only needed for specific functions that require a cosmology.
:param z_source: redshift of the source: Needed in multi_plane option only,
not required for the core functionalities in the single plane mode.
:param lens_redshift_list: list of deflector redshift (corresponding to the lens model list),
only applicable in multi_plane mode.
:param cosmo: instance of the astropy cosmology class. If not specified, uses the default cosmology.
:param multi_plane: bool, if True, uses multi-plane mode. Default is False.
:param numerical_alpha_class: an instance of a custom class for use in NumericalAlpha() lens model
(see documentation in Profiles/numerical_alpha)
:param observed_convention_index: a list of indices, corresponding to the lens_model_list element with same
index, where the 'center_x' and 'center_y' kwargs correspond to observed (lensed) positions, not physical
positions. The code will compute the physical locations when performing computations
:param z_source_convention: float, redshift of a source to define the reduced deflection angles of the lens
models. If None, 'z_source' is used.
:param cosmo_interp: boolean (only employed in multi-plane mode), interpolates astropy.cosmology distances for
faster calls when accessing several lensing planes
:param z_interp_stop: (only in multi-plane with cosmo_interp=True); maximum redshift for distance interpolation
This number should be higher or equal the maximum of the source redshift and/or the z_source_convention
:param num_z_interp: (only in multi-plane with cosmo_interp=True); number of redshift bins for interpolating
distances
"""
self.lens_model_list = lens_model_list
self.z_lens = z_lens
self.z_source = z_source
self._z_source_convention = z_source_convention
self.redshift_list = lens_redshift_list
if cosmo is None:
from astropy.cosmology import default_cosmology
cosmo = default_cosmology.get()
self.cosmo = cosmo
self.multi_plane = multi_plane
if multi_plane is True:
if z_source is None:
raise ValueError(
'z_source needs to be set for multi-plane lens modelling.')
self.lens_model = MultiPlane(
z_source,
lens_model_list,
lens_redshift_list,
cosmo=cosmo,
numerical_alpha_class=numerical_alpha_class,
observed_convention_index=observed_convention_index,
z_source_convention=z_source_convention,
cosmo_interp=cosmo_interp,
z_interp_stop=z_interp_stop,
num_z_interp=num_z_interp)
else:
self.lens_model = SinglePlane(
lens_model_list,
numerical_alpha_class=numerical_alpha_class,
lens_redshift_list=lens_redshift_list,
z_source_convention=z_source_convention)
if z_lens is not None and z_source is not None:
self._lensCosmo = LensCosmo(z_lens, z_source, cosmo=cosmo)
def ray_shooting(self, x, y, kwargs, k=None):
"""
maps image to source position (inverse deflection)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: source plane positions corresponding to (x, y) in the image plane
"""
return self.lens_model.ray_shooting(x, y, kwargs, k=k)
def fermat_potential(self,
x_image,
y_image,
kwargs_lens,
x_source=None,
y_source=None):
"""
fermat potential (negative sign means earlier arrival time)
for Multi-plane lensing, it computes the effective Fermat potential (derived from the arrival time and
subtracted off the time-delay distance for the given cosmology). The units are given in arcsecond square.
:param x_image: image position
:param y_image: image position
:param x_source: source position
:param y_source: source position
:param kwargs_lens: list of keyword arguments of lens model parameters matching the lens model classes
:return: fermat potential in arcsec**2 without geometry term (second part of Eqn 1 in Suyu et al. 2013) as a list
"""
if hasattr(self.lens_model, 'fermat_potential'):
return self.lens_model.fermat_potential(x_image, y_image,
kwargs_lens, x_source,
y_source)
elif hasattr(self.lens_model, 'arrival_time') and hasattr(
self, '_lensCosmo'):
dt = self.lens_model.arrival_time(x_image, y_image, kwargs_lens)
fermat_pot_eff = dt * const.c / self._lensCosmo.ddt / const.Mpc * const.day_s / const.arcsec**2
return fermat_pot_eff
else:
raise ValueError(
'In multi-plane lensing you need to provide a specific z_lens and z_source for which the '
'effective Fermat potential is evaluated')
def arrival_time(self, x_image, y_image, kwargs_lens, kappa_ext=0):
"""
:param x_image: image position
:param y_image: image position
:param kwargs_lens: lens model parameter keyword argument list
:param kappa_ext: external convergence contribution not accounted in the lens model that leads to the same
observables in position and relative fluxes but rescales the time delays
:return: arrival time of image positions in units of days
"""
if hasattr(self.lens_model, 'arrival_time'):
arrival_time = self.lens_model.arrival_time(
x_image, y_image, kwargs_lens)
else:
fermat_pot = self.lens_model.fermat_potential(
x_image, y_image, kwargs_lens)
if not hasattr(self, '_lensCosmo'):
raise ValueError(
"LensModel class was not initialized with lens and source redshifts!"
)
arrival_time = self._lensCosmo.time_delay_units(fermat_pot)
arrival_time *= (1 - kappa_ext)
return arrival_time
def potential(self, x, y, kwargs, k=None):
"""
lensing potential
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:return: lensing potential in units of arcsec^2
"""
return self.lens_model.potential(x, y, kwargs, k=k)
def alpha(self, x, y, kwargs, k=None, diff=None):
"""
deflection angles
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: None or float. If set, computes the deflection as a finite numerical differential of the lensing
potential. This differential is only applicable in the single lensing plane where the form of the lensing
potential is analytically known
:return: deflection angles in units of arcsec
"""
if diff is None:
return self.lens_model.alpha(x, y, kwargs, k=k)
elif self.multi_plane is False:
return self._deflection_differential(x, y, kwargs, k=k, diff=diff)
else:
raise ValueError(
'numerical differentiation of lensing potential is not available in the multi-plane '
'setting as analytical form of lensing potential is not available.'
)
def hessian(self, x, y, kwargs, k=None, diff=None, diff_method='square'):
"""
hessian matrix
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: float, scale over which the finite numerical differential is computed. If None, then using the
exact (if available) differentials.
:param diff_method: string, 'square' or 'cross', indicating whether finite differentials are computed from a
cross or a square of points around (x, y)
:return: f_xx, f_xy, f_yx, f_yy components
"""
if diff is None:
return self.lens_model.hessian(x, y, kwargs, k=k)
elif diff_method == 'square':
return self._hessian_differential_square(x,
y,
kwargs,
k=k,
diff=diff)
elif diff_method == 'cross':
return self._hessian_differential_cross(x,
y,
kwargs,
k=k,
diff=diff)
else:
raise ValueError(
'diff_method %s not supported. Chose among "square" or "cross".'
% diff_method)
def kappa(self, x, y, kwargs, k=None, diff=None, diff_method='square'):
"""
lensing convergence k = 1/2 laplacian(phi)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: float, scale over which the finite numerical differential is computed. If None, then using the
exact (if available) differentials.
:param diff_method: string, 'square' or 'cross', indicating whether finite differentials are computed from a
cross or a square of points around (x, y)
:return: lensing convergence
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x,
y,
kwargs,
k=k,
diff=diff,
diff_method=diff_method)
kappa = 1. / 2 * (f_xx + f_yy)
return kappa
def curl(self, x, y, kwargs, k=None, diff=None, diff_method='square'):
"""
curl computation F_xy - F_yx
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: float, scale over which the finite numerical differential is computed. If None, then using the
exact (if available) differentials.
:param diff_method: string, 'square' or 'cross', indicating whether finite differentials are computed from a
cross or a square of points around (x, y)
:return: curl at position (x, y)
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x,
y,
kwargs,
k=k,
diff=diff,
diff_method=diff_method)
return f_xy - f_yx
def gamma(self, x, y, kwargs, k=None, diff=None, diff_method='square'):
"""
shear computation
g1 = 1/2(d^2phi/dx^2 - d^2phi/dy^2)
g2 = d^2phi/dxdy
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: float, scale over which the finite numerical differential is computed. If None, then using the
exact (if available) differentials.
:param diff_method: string, 'square' or 'cross', indicating whether finite differentials are computed from a
cross or a square of points around (x, y)
:return: gamma1, gamma2
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x,
y,
kwargs,
k=k,
diff=diff,
diff_method=diff_method)
gamma1 = 1. / 2 * (f_xx - f_yy)
gamma2 = f_xy
return gamma1, gamma2
def magnification(self,
x,
y,
kwargs,
k=None,
diff=None,
diff_method='square'):
"""
magnification
mag = 1/det(A)
A = 1 - d^2phi/d_ij
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: only evaluate the k-th lens model
:param diff: float, scale over which the finite numerical differential is computed. If None, then using the
exact (if available) differentials.
:param diff_method: string, 'square' or 'cross', indicating whether finite differentials are computed from a
cross or a square of points around (x, y)
:return: magnification
"""
f_xx, f_xy, f_yx, f_yy = self.hessian(x,
y,
kwargs,
k=k,
diff=diff,
diff_method=diff_method)
det_A = (1 - f_xx) * (1 - f_yy) - f_xy * f_yx
return 1. / det_A # attention, if dividing by zero
def flexion(self, x, y, kwargs, k=None, diff=0.000001, hessian_diff=True):
"""
third derivatives (flexion)
:param x: x-position (preferentially arcsec)
:type x: numpy array
:param y: y-position (preferentially arcsec)
:type y: numpy array
:param kwargs: list of keyword arguments of lens model parameters matching the lens model classes
:param k: int or None, if set, only evaluates the differential from one model component
:param diff: numerical differential length of Flexion
:param hessian_diff: boolean, if true also computes the numerical differential length of Hessian (optional)
:return: f_xxx, f_xxy, f_xyy, f_yyy
"""
if hessian_diff is not True:
hessian_diff = None
f_xx_dx, f_xy_dx, f_yx_dx, f_yy_dx = self.hessian(x + diff / 2,
y,
kwargs,
k=k,
diff=hessian_diff)
f_xx_dy, f_xy_dy, f_yx_dy, f_yy_dy = self.hessian(x,
y + diff / 2,
kwargs,
k=k,
diff=hessian_diff)
f_xx_dx_, f_xy_dx_, f_yx_dx_, f_yy_dx_ = self.hessian(
x - diff / 2, y, kwargs, k=k, diff=hessian_diff)
f_xx_dy_, f_xy_dy_, f_yx_dy_, f_yy_dy_ = self.hessian(
x, y - diff / 2, kwargs, k=k, diff=hessian_diff)
f_xxx = (f_xx_dx - f_xx_dx_) / diff
f_xxy = (f_xx_dy - f_xx_dy_) / diff
f_xyy = (f_xy_dy - f_xy_dy_) / diff
f_yyy = (f_yy_dy - f_yy_dy_) / diff
return f_xxx, f_xxy, f_xyy, f_yyy
def set_static(self, kwargs):
"""
set this instance to a static lens model. This can improve the speed in evaluating lensing quantities at
different positions but must not be used with different lens model parameters!
:param kwargs: lens model keyword argument list
:return: kwargs_updated (in case of image position convention in multiplane lensing this is changed)
"""
return self.lens_model.set_static(kwargs)
def set_dynamic(self):
"""
deletes cache for static setting and makes sure the observed convention in the position of lensing profiles in
the multi-plane setting is enabled. Dynamic is the default setting of this class enabling an accurate computation
of lensing quantities with different parameters in the lensing profiles.
:return: None
"""
self.lens_model.set_dynamic()
def _deflection_differential(self, x, y, kwargs, k=None, diff=0.00001):
"""
:param x: x-coordinate
:param y: y-coordinate
:param kwargs: keyword argument list
:param k: int or None, if set, only evaluates the differential from one model component
:param diff: finite differential length
:return: f_x, f_y
"""
phi_dx = self.lens_model.potential(x + diff / 2, y, kwargs=kwargs, k=k)
phi_dy = self.lens_model.potential(x, y + diff / 2, kwargs=kwargs, k=k)
phi_dx_ = self.lens_model.potential(x - diff / 2,
y,
kwargs=kwargs,
k=k)
phi_dy_ = self.lens_model.potential(x,
y - diff / 2,
kwargs=kwargs,
k=k)
f_x = (phi_dx - phi_dx_) / diff
f_y = (phi_dy - phi_dy_) / diff
return f_x, f_y
def _hessian_differential_cross(self, x, y, kwargs, k=None, diff=0.00001):
"""
computes the numerical differentials over a finite range for f_xx, f_yy, f_xy from f_x and f_y
The differentials are computed along the cross centered at (x, y).
:param x: x-coordinate
:param y: y-coordinate
:param kwargs: lens model keyword argument list
:param k: int, list of bools or None, indicating a subset of lens models to be evaluated
:param diff: float, scale of the finite differential (diff/2 in each direction used to compute the differential
:return: f_xx, f_xy, f_yx, f_yy
"""
alpha_ra_dx, alpha_dec_dx = self.alpha(x + diff / 2, y, kwargs, k=k)
alpha_ra_dy, alpha_dec_dy = self.alpha(x, y + diff / 2, kwargs, k=k)
alpha_ra_dx_, alpha_dec_dx_ = self.alpha(x - diff / 2, y, kwargs, k=k)
alpha_ra_dy_, alpha_dec_dy_ = self.alpha(x, y - diff / 2, kwargs, k=k)
dalpha_rara = (alpha_ra_dx - alpha_ra_dx_) / diff
dalpha_radec = (alpha_ra_dy - alpha_ra_dy_) / diff
dalpha_decra = (alpha_dec_dx - alpha_dec_dx_) / diff
dalpha_decdec = (alpha_dec_dy - alpha_dec_dy_) / diff
f_xx = dalpha_rara
f_yy = dalpha_decdec
f_xy = dalpha_radec
f_yx = dalpha_decra
return f_xx, f_xy, f_yx, f_yy
def _hessian_differential_square(self, x, y, kwargs, k=None, diff=0.00001):
"""
computes the numerical differentials over a finite range for f_xx, f_yy, f_xy from f_x and f_y
The differentials are computed on the square around (x, y). This minimizes curl.
:param x: x-coordinate
:param y: y-coordinate
:param kwargs: lens model keyword argument list
:param k: int, list of booleans or None, indicating a subset of lens models to be evaluated
:param diff: float, scale of the finite differential (diff/2 in each direction used to compute the differential
:return: f_xx, f_xy, f_yx, f_yy
"""
alpha_ra_pp, alpha_dec_pp = self.alpha(x + diff / 2,
y + diff / 2,
kwargs,
k=k)
alpha_ra_pn, alpha_dec_pn = self.alpha(x + diff / 2,
y - diff / 2,
kwargs,
k=k)
alpha_ra_np, alpha_dec_np = self.alpha(x - diff / 2,
y + diff / 2,
kwargs,
k=k)
alpha_ra_nn, alpha_dec_nn = self.alpha(x - diff / 2,
y - diff / 2,
kwargs,
k=k)
f_xx = (alpha_ra_pp - alpha_ra_np + alpha_ra_pn -
alpha_ra_nn) / diff / 2
f_xy = (alpha_ra_pp - alpha_ra_pn + alpha_ra_np -
alpha_ra_nn) / diff / 2
f_yx = (alpha_dec_pp - alpha_dec_np + alpha_dec_pn -
alpha_dec_nn) / diff / 2
f_yy = (alpha_dec_pp - alpha_dec_pn + alpha_dec_np -
alpha_dec_nn) / diff / 2
return f_xx, f_xy, f_yx, f_yy