__author__ = 'sibirrer'
# description of the polar shapelets in potential space
import numpy as np
import scipy.special
import math
import lenstronomy.Util.param_util as param_util
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ['PolarShapelets']
[docs]class PolarShapelets(LensProfileBase):
"""
this class contains the function and the derivatives of the Singular Isothermal Sphere
"""
param_names = ['coeffs', 'beta', 'center_x', 'center_y']
lower_limit_default = {'coeffs': [0], 'beta': 0, 'center_x': -100, 'center_y': -100}
upper_limit_default = {'coeffs': [100], 'beta': 100, 'center_x': 100, 'center_y': 100}
def __init__(self):
n = 10
self.poly = [[[] for i in range(n)] for j in range(n)]
for i in range(0, n):
for j in range(0, n):
self.poly[i][j] = scipy.special.genlaguerre(i, j)
super(PolarShapelets, self).__init__()
[docs] def function(self, x, y, coeffs, beta, center_x=0, center_y=0):
shapelets = self._createShapelet(coeffs)
r, phi = param_util.cart2polar(x, y, center_x=center_x, center_y=center_y)
f_ = self._shapeletOutput(r, phi, beta, shapelets)
return f_
[docs] def derivatives(self, x, y, coeffs, beta, center_x=0, center_y=0):
"""
returns df/dx and df/dy of the function
"""
shapelets = self._createShapelet(coeffs)
r, phi = param_util.cart2polar(x, y, center_x=center_x, center_y=center_y)
alpha1_shapelets, alpha2_shapelets = self._alphaShapelets(shapelets, beta)
f_x = self._shapeletOutput(r, phi, beta, alpha1_shapelets)
f_y = self._shapeletOutput(r, phi, beta, alpha2_shapelets)
return f_x, f_y
[docs] def hessian(self, x, y, coeffs, beta, center_x=0, center_y=0):
"""
returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
"""
shapelets = self._createShapelet(coeffs)
r, phi = param_util.cart2polar(x, y, center_x=center_x, center_y=center_y)
kappa_shapelets=self._kappaShapelets(shapelets, beta)
gamma1_shapelets, gamma2_shapelets=self._gammaShapelets(shapelets, beta)
kappa_value=self._shapeletOutput(r, phi, beta, kappa_shapelets)
gamma1_value=self._shapeletOutput(r, phi, beta, gamma1_shapelets)
gamma2_value=self._shapeletOutput(r, phi, beta, gamma2_shapelets)
f_xx = kappa_value + gamma1_value
f_xy = gamma2_value
f_yy = kappa_value - gamma1_value
return f_xx, f_xy, f_xy, f_yy
def _createShapelet(self,coeff):
"""
returns a shapelet array out of the coefficients *a, up to order l
:param num_l: order of shapelets
:type num_l: int.
:param coeff: shapelet coefficients
:type coeff: floats
:returns: complex array
:raises: AttributeError, KeyError
"""
n_coeffs = len(coeff)
num_l = self._get_num_l(n_coeffs)
shapelets=np.zeros((num_l+1,num_l+1),'complex')
nl=0
k=0
i=0
while i < len(coeff):
if i%2==0:
shapelets[nl][k]+=coeff[i]/2.
shapelets[k][nl]+=coeff[i]/2.
if k==nl:
nl+=1
k=0
i+=1
continue
else:
k+=1
i+=1
continue
else:
shapelets[nl][k] += 1j*coeff[i]/2.
shapelets[k][nl] -= 1j*coeff[i]/2.
i+=1
return shapelets
def _shapeletOutput(self, r, phi, beta, shapelets):
"""
returns the the numerical values of a set of shapelets at polar coordinates
:param shapelets: set of shapelets [l=,r=,a_lr=]
:type shapelets: array of size (n,3)
:param coordPolar: set of coordinates in polar units
:type coordPolar: array of size (n,2)
:returns: array of same size with coords [r,phi]
:raises: AttributeError, KeyError
"""
if type(r) == float or type(r) == int or type(r) == type(np.float64(1)) or len(r) <= 1:
values = 0.
else:
values = np.zeros(len(r), 'complex')
for nl in range(0,len(shapelets)): #sum over different shapelets
for nr in range(0,len(shapelets)):
value = shapelets[nl][nr]*self._chi_lr(r, phi, nl, nr, beta)
values += value
return values.real
def _chi_lr(self, r, phi, nl, nr, beta):
"""
computes the generalized polar basis function in the convention of Massey&Refregier eqn 8
:param nl: left basis
:type nl: int
:param nr: right basis
:type nr: int
:param beta: beta --the characteristic scale typically choosen to be close to the size of the object.
:type beta: float.
:param coord: coordinates [r,phi]
:type coord: array(n,2)
:returns: values at positions of coordinates.
:raises: AttributeError, KeyError
"""
m=int((nr-nl).real)
n=int((nr+nl).real)
p=int((n-abs(m))/2)
p2=int((n+abs(m))/2)
q=int(abs(m))
if p % 2==0: #if p is even
prefac=1
else:
prefac=-1
prefactor=prefac/beta**(abs(m)+1)*np.sqrt(math.factorial(p)/(np.pi*math.factorial(p2)))
poly=self.poly[p][q]
return prefactor*r**q*poly((r/beta)**2)*np.exp(-(r/beta)**2/2)*np.exp(-1j*m*phi)
def _kappaShapelets(self, shapelets, beta):
"""
calculates the convergence kappa given lensing potential shapelet coefficients (laplacian/2)
:param shapelets: set of shapelets [l=,r=,a_lr=]
:type shapelets: array of size (n,3)
:returns: set of kappa shapelets.
:raises: AttributeError, KeyError
"""
output=np.zeros((len(shapelets)+1,len(shapelets)+1),'complex')
for nl in range(0,len(shapelets)):
for nr in range(0,len(shapelets)):
a_lr=shapelets[nl][nr]
if nl>0:
output[nl-1][nr+1]+=a_lr*np.sqrt(nl*(nr+1))/2
if nr>0:
output[nl-1][nr-1]+=a_lr*np.sqrt(nl*nr)/2
output[nl+1][nr+1]+=a_lr*np.sqrt((nl+1)*(nr+1))/2
if nr>0:
output[nl+1][nr-1]+=a_lr*np.sqrt((nl+1)*nr)/2
return output/beta**2
def _alphaShapelets(self,shapelets, beta):
"""
calculates the deflection angles given lensing potential shapelet coefficients (laplacian/2)
:param shapelets: set of shapelets [l=,r=,a_lr=]
:type shapelets: array of size (n,3)
:returns: set of alpha shapelets.
:raises: AttributeError, KeyError
"""
output_x = np.zeros((len(shapelets)+1, len(shapelets)+1), 'complex')
output_y = np.zeros((len(shapelets)+1, len(shapelets)+1), 'complex')
for nl in range(0,len(shapelets)):
for nr in range(0,len(shapelets)):
a_lr=shapelets[nl][nr]
output_x[nl][nr+1]-=a_lr*np.sqrt(nr+1)/2
output_y[nl][nr+1]-=a_lr*np.sqrt(nr+1)/2*1j
output_x[nl+1][nr]-=a_lr*np.sqrt(nl+1)/2
output_y[nl+1][nr]+=a_lr*np.sqrt(nl+1)/2*1j
if nl>0:
output_x[nl-1][nr]+=a_lr*np.sqrt(nl)/2
output_y[nl-1][nr]-=a_lr*np.sqrt(nl)/2*1j
if nr>0:
output_x[nl][nr-1]+=a_lr*np.sqrt(nr)/2
output_y[nl][nr-1]+=a_lr*np.sqrt(nr)/2*1j
return output_x/beta,output_y/beta #attention complex numbers!!!!
def _gammaShapelets(self,shapelets, beta):
"""
calculates the shear gamma given lensing potential shapelet coefficients
:param shapelets: set of shapelets [l=,r=,a_lr=]
:type shapelets: array of size (n,3)
:returns: set of alpha shapelets.
:raises: AttributeError, KeyError
"""
output_x = np.zeros((len(shapelets)+2,len(shapelets)+2),'complex')
output_y = np.zeros((len(shapelets)+2,len(shapelets)+2),'complex')
for nl in range(0, len(shapelets)):
for nr in range(0, len(shapelets)):
a_lr = shapelets[nl][nr]
output_x[nl+2][nr] += a_lr*np.sqrt((nl+1)*(nl+2))/2
output_x[nl][nr+2] += a_lr*np.sqrt((nr+1)*(nr+2))/2
output_x[nl][nr] += a_lr*(1-(nr+1)-(nl+1))
if nl>1:
output_x[nl-2][nr] += a_lr*np.sqrt((nl)*(nl-1))/2
if nr>1:
output_x[nl][nr-2] += a_lr*np.sqrt((nr)*(nr-1))/2
output_y[nl+2][nr] += a_lr*np.sqrt((nl+1)*(nl+2))*1j/4
output_y[nl][nr+2] -= a_lr*np.sqrt((nr+1)*(nr+2))*1j/4
if nl>0:
output_y[nl-1][nr+1] += a_lr*np.sqrt((nl)*(nr+1))*1j/2
if nr>0:
output_y[nl+1][nr-1] -= a_lr*np.sqrt((nr)*(nl+1))*1j/2
if nl>1:
output_y[nl-2][nr] -= a_lr*np.sqrt((nl)*(nl-1))*1j/4
if nr>1:
output_y[nl][nr-2] += a_lr*np.sqrt((nr)*(nr-1))*1j/4
return output_x/beta**2, output_y/beta**2 #attention complex numbers!!!!
@staticmethod
def _get_num_l(n_coeffs):
"""
:param n_coeffs: number of coeffs
:return: number of n_l of order of the shapelets
"""
num_l = int(round((math.sqrt(8*n_coeffs + 9)-3)/2 +0.499))
return num_l