class GaussianEllipsePotential(LensProfileBase):
"""
this class contains functions to evaluate a Gaussian function and calculates its derivative and hessian matrix
with ellipticity in the convergence
the calculation follows Glenn van de Ven et al. 2009
"""
param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']
lower_limit_default = {
'amp': 0,
'sigma': 0,
'e1': -0.5,
'e2': -0.5,
'center_x': -100,
'center_y': -100
}
upper_limit_default = {
'amp': 100,
'sigma': 100,
'e1': 0.5,
'e2': 0.5,
'center_x': 100,
'center_y': 100
}
def __init__(self):
self.spherical = GaussianKappa()
self._diff = 0.000001
super(GaussianEllipsePotential, self).__init__()
def function(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
returns Gaussian
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = abs(1 - q)
x_ = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
y_ = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
f_ = self.spherical.function(x_, y_, amp=amp, sigma=sigma)
return f_
def derivatives(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
returns df/dx and df/dy of the function
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
x_shift = x - center_x
y_shift = y - center_y
cos_phi = np.cos(phi_G)
sin_phi = np.sin(phi_G)
e = abs(1 - q)
x_ = (cos_phi * x_shift + sin_phi * y_shift) * np.sqrt(1 - e)
y_ = (-sin_phi * x_shift + cos_phi * y_shift) * np.sqrt(1 + e)
f_x_prim, f_y_prim = self.spherical.derivatives(x_,
y_,
amp=amp,
sigma=sigma)
f_x_prim *= np.sqrt(1 - e)
f_y_prim *= np.sqrt(1 + e)
f_x = cos_phi * f_x_prim - sin_phi * f_y_prim
f_y = sin_phi * f_x_prim + cos_phi * f_y_prim
return f_x, f_y
def hessian(self, x, y, amp, sigma, e1, e2, 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
"""
alpha_ra, alpha_dec = self.derivatives(x, y, amp, sigma, e1, e2,
center_x, center_y)
diff = self._diff
alpha_ra_dx, alpha_dec_dx = self.derivatives(x + diff, y, amp, sigma,
e1, e2, center_x,
center_y)
alpha_ra_dy, alpha_dec_dy = self.derivatives(x, y + diff, amp, sigma,
e1, e2, center_x,
center_y)
f_xx = (alpha_ra_dx - alpha_ra) / diff
f_xy = (alpha_ra_dy - alpha_ra) / diff
f_yx = (alpha_dec_dx - alpha_dec) / diff
f_yy = (alpha_dec_dy - alpha_dec) / diff
return f_xx, f_xy, f_yx, f_yy
def density(self, r, amp, sigma, e1, e2):
"""
:param r:
:param amp:
:param sigma:
:return:
"""
return self.spherical.density(r, amp, sigma)
def density_2d(self, x, y, amp, sigma, e1, e2, center_x=0, center_y=0):
"""
:param R:
:param am:
:param sigma_x:
:param sigma_y:
:return:
"""
return self.spherical.density_2d(x, y, amp, sigma, center_x, center_y)
def mass_2d(self, R, amp, sigma, e1, e2):
"""
:param R:
:param amp:
:param sigma_x:
:param sigma_y:
:return:
"""
return self.spherical.mass_2d(R, amp, sigma)
def mass_3d(self, R, amp, sigma, e1, e2):
"""
:param R:
:param amp:
:param sigma:
:param e1:
:param e2:
:return:
"""
return self.spherical.mass_3d(R, amp, sigma)
def mass_3d_lens(self, R, amp, sigma, e1, e2):
"""
:param R:
:param amp:
:param sigma:
:param e1:
:param e2:
:return:
"""
return self.spherical.mass_3d_lens(R, amp, sigma)
def mass_2d_lens(self, R, amp, sigma, e1, e2):
"""
:param R:
:param amp:
:param sigma_x:
:param sigma_y:
:return:
"""
return self.spherical.mass_2d_lens(R, amp, sigma)
class TestGaussianKappaPot(object):
"""
test the Gaussian with Gaussian kappa
"""
def setup(self):
self.gaussian_kappa = GaussianKappa()
self.ellipse = GaussianEllipsePotential()
def test_function(self):
x = 1
y = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_ = self.ellipse.function(x, y, amp, sigma, e1, e2)
f_sphere = self.gaussian_kappa.function(x, y, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_, f_sphere, decimal=8)
def test_derivatives(self):
x = 1
y = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_x, f_y = self.ellipse.derivatives(x, y, amp, sigma, e1, e2)
f_x_sphere, f_y_sphere = self.gaussian_kappa.derivatives(x, y, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_x, f_x_sphere, decimal=8)
npt.assert_almost_equal(f_y, f_y_sphere, decimal=8)
def test_hessian(self):
x = 1
y = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_xx, f_xy, f_yx, f_yy = self.ellipse.hessian(x, y, amp, sigma, e1, e2)
f_xx_sphere, f_xy_sphere, f_yx_sphere, f_yy_sphere = self.gaussian_kappa.hessian(x, y, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_xx, f_xx_sphere, decimal=5)
npt.assert_almost_equal(f_yy, f_yy_sphere, decimal=5)
npt.assert_almost_equal(f_xy, f_xy_sphere, decimal=5)
npt.assert_almost_equal(f_xy, f_yx, decimal=8)
def test_density_2d(self):
x = 1
y = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_ = self.ellipse.density_2d(x, y, amp, sigma, e1, e2)
f_sphere = self.gaussian_kappa.density_2d(x, y, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_, f_sphere, decimal=8)
def test_mass_2d(self):
r = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_ = self.ellipse.mass_2d(r, amp, sigma, e1, e2)
f_sphere = self.gaussian_kappa.mass_2d(r, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_, f_sphere, decimal=8)
def test_mass_2d_lens(self):
r = 1
e1, e2 = 0, 0
sigma = 1
amp = 1
f_ = self.ellipse.mass_2d_lens(r, amp, sigma, e1, e2)
f_sphere = self.gaussian_kappa.mass_2d_lens(r, amp=amp, sigma=sigma)
npt.assert_almost_equal(f_, f_sphere, decimal=8)
