def _Rzderiv(self, R, z, phi=0., t=0.):
"""
NAME:
_Rzderiv
PURPOSE:
evaluate the mixed R,z derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t- time
OUTPUT:
d2phi/dR/dz
HISTORY:
2015-02-13 - Written - Trick (MPIA)
"""
l, n = bovy_coords.Rz_to_lambdanu(R, z, ac=self._ac, Delta=self._Delta)
jac = bovy_coords.Rz_to_lambdanu_jac(R, z, Delta=self._Delta)
hess = bovy_coords.Rz_to_lambdanu_hess(R, z, Delta=self._Delta)
dldR = jac[0, 0]
dndR = jac[1, 0]
dldz = jac[0, 1]
dndz = jac[1, 1]
d2ldRdz = hess[0, 0, 1]
d2ndRdz = hess[1, 0, 1]
return d2ldRdz * self._lderiv(l,n) + \
d2ndRdz * self._nderiv(l,n) + \
dldR*dldz * self._l2deriv(l,n) + \
dndR*dndz * self._n2deriv(l,n) + \
(dldR*dndz+dldz*dndR)* self._lnderiv(l,n)
def test_Rz_to_lambdanu_r2lt0():
# Special case that leads to r2 just below zero
#coordinate system:
a = 3.
g = 7.
Delta = numpy.sqrt(g - a)
ac = numpy.sqrt(a / g)
#_____test float input (R=0)_____
#true values:
l, n = 2., -3. + 10.**-10.
#coordinate transformation:
lt, nt = bovy_coords.Rz_to_lambdanu(*bovy_coords.lambdanu_to_Rz(
l, n, ac=ac, Delta=Delta),
ac=ac,
Delta=Delta)
#test:
assert numpy.fabs(
lt -
l) < 10.**-8., 'Rz_to_lambdanu conversion did not work as expected (l)'
assert numpy.fabs(
nt -
n) < 10.**-8., 'Rz_to_lambdanu conversion did not work as expected (n)'
#___Also test for arrays___
l = numpy.array([2., 10., 20., 0.])
n = numpy.array([-7., -3. + 10.**-10., -5., -5.])
lt, nt = bovy_coords.Rz_to_lambdanu(*bovy_coords.lambdanu_to_Rz(
l, n, ac=ac, Delta=Delta),
ac=ac,
Delta=Delta)
assert numpy.all(
numpy.fabs(lt - l) < 10.**-8.
), 'Rz_to_lambdanu conversion did not work as expected (l array)'
assert numpy.all(
numpy.fabs(nt - n) < 10.**-8.
), 'Rz_to_lambdanu conversion did not work as expected (n array)'
return None
def _evaluate(self, R, z, phi=0., t=0.):
"""
NAME:
_evaluate
PURPOSE:
evaluate the potential at R,z
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
Phi(R,z)
HISTORY:
2015-02-15 - Written - Trick (MPIA)
"""
l, n = bovy_coords.Rz_to_lambdanu(R, z, ac=self._ac, Delta=self._Delta)
return -1. / (nu.sqrt(l) + nu.sqrt(n))
def _Rforce(self, R, z, phi=0., t=0.):
"""
NAME:
_Rforce
PURPOSE:
evaluate the radial force for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the radial force = -dphi/dR
HISTORY:
2015-02-13 - Written - Trick (MPIA)
"""
l, n = bovy_coords.Rz_to_lambdanu(R, z, ac=self._ac, Delta=self._Delta)
jac = bovy_coords.Rz_to_lambdanu_jac(R, z, Delta=self._Delta)
dldR = jac[0, 0]
dndR = jac[1, 0]
return - (dldR * self._lderiv(l,n) + \
dndR * self._nderiv(l,n))
def test_Rz_to_lambdanu_hess():
#coordinate system:
a = 3.
g = 7.
Delta = numpy.sqrt(g - a)
ac = numpy.sqrt(a / g)
#_____test float input (R=0)_____
R, z = 1.4, 0.1
dR = 10.**-5.
dz = 10.**-5.
# R derivatives
tmp = R + dR
dR = tmp - R
# z derivatives
tmp = z + dz
dz = tmp - z
num_deriv_llRR = (
bovy_coords.Rz_to_lambdanu(R + dR, z, ac=ac, Delta=Delta)[0] -
2. * bovy_coords.Rz_to_lambdanu(R, z, ac=ac, Delta=Delta)[0] +
bovy_coords.Rz_to_lambdanu(R - dR, z, ac=ac, Delta=Delta)[0]) / dR**2.
num_deriv_nnRR = (
bovy_coords.Rz_to_lambdanu(R + dR, z, ac=ac, Delta=Delta)[1] -
2. * bovy_coords.Rz_to_lambdanu(R, z, ac=ac, Delta=Delta)[1] +
bovy_coords.Rz_to_lambdanu(R - dR, z, ac=ac, Delta=Delta)[1]) / dR**2.
num_deriv_llRz = (
bovy_coords.Rz_to_lambdanu(R + dR, z + dz, ac=ac, Delta=Delta)[0] -
bovy_coords.Rz_to_lambdanu(R + dR, z - dz, ac=ac, Delta=Delta)[0] -
bovy_coords.Rz_to_lambdanu(R - dR, z + dz, ac=ac, Delta=Delta)[0] +
bovy_coords.Rz_to_lambdanu(R - dR, z - dz, ac=ac,
Delta=Delta)[0]) / dR**2. / 4.
num_deriv_llzz = (
bovy_coords.Rz_to_lambdanu(R, z + dz, ac=ac, Delta=Delta)[0] -
2. * bovy_coords.Rz_to_lambdanu(R, z, ac=ac, Delta=Delta)[0] +
bovy_coords.Rz_to_lambdanu(R, z - dz, ac=ac, Delta=Delta)[0]) / dz**2.
num_deriv_nnzz = (
bovy_coords.Rz_to_lambdanu(R, z + dz, ac=ac, Delta=Delta)[1] -
2. * bovy_coords.Rz_to_lambdanu(R, z, ac=ac, Delta=Delta)[1] +
bovy_coords.Rz_to_lambdanu(R, z - dz, ac=ac, Delta=Delta)[1]) / dz**2.
num_deriv_nnRz = (
bovy_coords.Rz_to_lambdanu(R + dR, z + dz, ac=ac, Delta=Delta)[1] -
bovy_coords.Rz_to_lambdanu(R + dR, z - dz, ac=ac, Delta=Delta)[1] -
bovy_coords.Rz_to_lambdanu(R - dR, z + dz, ac=ac, Delta=Delta)[1] +
bovy_coords.Rz_to_lambdanu(R - dR, z - dz, ac=ac,
Delta=Delta)[1]) / dR**2. / 4.
hess = bovy_coords.Rz_to_lambdanu_hess(R, z, Delta=Delta)
assert numpy.fabs(
num_deriv_llRR - hess[0, 0, 0]
) < 10.**-4., 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dl/dR)'
assert numpy.fabs(
num_deriv_llRz - hess[0, 0, 1]
) < 10.**-4., 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dn/dR)'
assert numpy.fabs(
num_deriv_nnRR - hess[1, 0, 0]
) < 10.**-4., 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dn/dR)'
assert numpy.fabs(
num_deriv_llzz - hess[0, 1, 1]
) < 10.**-4., 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dl/dz)'
assert numpy.fabs(
num_deriv_nnRz - hess[1, 0, 1]
) < 10.**-4., 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dn/dz)'
assert numpy.fabs(
num_deriv_nnzz - hess[1, 1, 1]
) < 10.**-4., 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dn/dz)'
#___Also test for arrays___
R = numpy.arange(1, 4) * 0.5
z = numpy.arange(1, 4) * 0.125
# R derivatives
tmp = R + dR
dR = tmp - R
# z derivatives
tmp = z + dz
dz = tmp - z
dR = 10.**-5.
dz = 10.**-5.
num_deriv_llRR = (
bovy_coords.Rz_to_lambdanu(R + dR, z, ac=ac, Delta=Delta)[0] -
2. * bovy_coords.Rz_to_lambdanu(R, z, ac=ac, Delta=Delta)[0] +
bovy_coords.Rz_to_lambdanu(R - dR, z, ac=ac, Delta=Delta)[0]) / dR**2.
num_deriv_nnRR = (
bovy_coords.Rz_to_lambdanu(R + dR, z, ac=ac, Delta=Delta)[1] -
2. * bovy_coords.Rz_to_lambdanu(R, z, ac=ac, Delta=Delta)[1] +
bovy_coords.Rz_to_lambdanu(R - dR, z, ac=ac, Delta=Delta)[1]) / dR**2.
num_deriv_llRz = (
bovy_coords.Rz_to_lambdanu(R + dR, z + dz, ac=ac, Delta=Delta)[0] -
bovy_coords.Rz_to_lambdanu(R + dR, z - dz, ac=ac, Delta=Delta)[0] -
bovy_coords.Rz_to_lambdanu(R - dR, z + dz, ac=ac, Delta=Delta)[0] +
bovy_coords.Rz_to_lambdanu(R - dR, z - dz, ac=ac,
Delta=Delta)[0]) / dR**2. / 4.
num_deriv_llzz = (
bovy_coords.Rz_to_lambdanu(R, z + dz, ac=ac, Delta=Delta)[0] -
2. * bovy_coords.Rz_to_lambdanu(R, z, ac=ac, Delta=Delta)[0] +
bovy_coords.Rz_to_lambdanu(R, z - dz, ac=ac, Delta=Delta)[0]) / dz**2.
num_deriv_nnzz = (
bovy_coords.Rz_to_lambdanu(R, z + dz, ac=ac, Delta=Delta)[1] -
2. * bovy_coords.Rz_to_lambdanu(R, z, ac=ac, Delta=Delta)[1] +
bovy_coords.Rz_to_lambdanu(R, z - dz, ac=ac, Delta=Delta)[1]) / dz**2.
num_deriv_nnRz = (
bovy_coords.Rz_to_lambdanu(R + dR, z + dz, ac=ac, Delta=Delta)[1] -
bovy_coords.Rz_to_lambdanu(R + dR, z - dz, ac=ac, Delta=Delta)[1] -
bovy_coords.Rz_to_lambdanu(R - dR, z + dz, ac=ac, Delta=Delta)[1] +
bovy_coords.Rz_to_lambdanu(R - dR, z - dz, ac=ac,
Delta=Delta)[1]) / dR**2. / 4.
hess = bovy_coords.Rz_to_lambdanu_hess(R, z, Delta=Delta)
assert numpy.all(
numpy.fabs(num_deriv_llRR - hess[0, 0, 0]) < 10.**-4.
), 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dl/dR)'
assert numpy.all(
numpy.fabs(num_deriv_llRz - hess[0, 0, 1]) < 10.**-4.
), 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dn/dR)'
assert numpy.all(
numpy.fabs(num_deriv_nnRR - hess[1, 0, 0]) < 10.**-4.
), 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dn/dR)'
assert numpy.all(
numpy.fabs(num_deriv_llzz - hess[0, 1, 1]) < 10.**-4.
), 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dl/dz)'
assert numpy.all(
numpy.fabs(num_deriv_nnRz - hess[1, 0, 1]) < 10.**-4.
), 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dn/dz)'
assert numpy.all(
numpy.fabs(num_deriv_nnzz - hess[1, 1, 1]) < 10.**-4.
), 'hessian [d^2(lamda)/d(R,z)^2 , d^2(nu)/d(R,z)^2] fails for (dn/dz)'
return None
def test_Rz_to_lambdanu_jac():
#coordinate system:
a = 3.
g = 7.
Delta = numpy.sqrt(g - a)
ac = numpy.sqrt(a / g)
#_____test float input (R=0)_____
R, z = 1.4, 0.1
dR = 10.**-8.
dz = 10.**-8.
# R derivatives
tmp = R + dR
dR = tmp - R
num_deriv_lR= (bovy_coords.Rz_to_lambdanu(R+dR,z,ac=ac,Delta=Delta)[0]\
-bovy_coords.Rz_to_lambdanu(R,z,ac=ac,Delta=Delta)[0])/dR
num_deriv_nR= (bovy_coords.Rz_to_lambdanu(R+dR,z,ac=ac,Delta=Delta)[1]\
-bovy_coords.Rz_to_lambdanu(R,z,ac=ac,Delta=Delta)[1])/dR
# z derivatives
tmp = z + dz
dz = tmp - z
num_deriv_lz= (bovy_coords.Rz_to_lambdanu(R,z+dz,ac=ac,Delta=Delta)[0]\
-bovy_coords.Rz_to_lambdanu(R,z,ac=ac,Delta=Delta)[0])/dR
num_deriv_nz= (bovy_coords.Rz_to_lambdanu(R,z+dz,ac=ac,Delta=Delta)[1]\
-bovy_coords.Rz_to_lambdanu(R,z,ac=ac,Delta=Delta)[1])/dR
jac = bovy_coords.Rz_to_lambdanu_jac(R, z, Delta=Delta)
assert numpy.fabs(num_deriv_lR - jac[
0, 0]) < 10.**-6., 'jacobian d((lambda,nu))/d((R,z)) fails for (dl/dR)'
assert numpy.fabs(num_deriv_nR - jac[
1, 0]) < 10.**-6., 'jacobian d((lambda,nu))/d((R,z)) fails for (dn/dR)'
assert numpy.fabs(num_deriv_lz - jac[
0, 1]) < 10.**-6., 'jacobian d((lambda,nu))/d((R,z)) fails for (dl/dz)'
assert numpy.fabs(num_deriv_nz - jac[
1, 1]) < 10.**-6., 'jacobian d((lambda,nu))/d((R,z)) fails for (dn/dz)'
#___Also test for arrays___
R = numpy.arange(1, 4) * 0.5
z = numpy.arange(1, 4) * 0.125
dR = 10.**-8.
dz = 10.**-8.
# R derivatives
tmp = R + dR
dR = tmp - R
num_deriv_lR= (bovy_coords.Rz_to_lambdanu(R+dR,z,ac=ac,Delta=Delta)[0]\
-bovy_coords.Rz_to_lambdanu(R,z,ac=ac,Delta=Delta)[0])/dR
num_deriv_nR= (bovy_coords.Rz_to_lambdanu(R+dR,z,ac=ac,Delta=Delta)[1]\
-bovy_coords.Rz_to_lambdanu(R,z,ac=ac,Delta=Delta)[1])/dR
# z derivatives
tmp = z + dz
dz = tmp - z
num_deriv_lz= (bovy_coords.Rz_to_lambdanu(R,z+dz,ac=ac,Delta=Delta)[0]\
-bovy_coords.Rz_to_lambdanu(R,z,ac=ac,Delta=Delta)[0])/dR
num_deriv_nz= (bovy_coords.Rz_to_lambdanu(R,z+dz,ac=ac,Delta=Delta)[1]\
-bovy_coords.Rz_to_lambdanu(R,z,ac=ac,Delta=Delta)[1])/dR
jac = bovy_coords.Rz_to_lambdanu_jac(R, z, Delta=Delta)
assert numpy.all(numpy.fabs(num_deriv_lR - jac[0, 0]) < 10.**-6.
), 'jacobian d((lambda,nu))/d((R,z)) fails for (dl/dR)'
assert numpy.all(numpy.fabs(num_deriv_nR - jac[1, 0]) < 10.**-6.
), 'jacobian d((lambda,nu))/d((R,z)) fails for (dn/dR)'
assert numpy.all(numpy.fabs(num_deriv_lz - jac[0, 1]) < 10.**-6.
), 'jacobian d((lambda,nu))/d((R,z)) fails for (dl/dz)'
assert numpy.all(numpy.fabs(num_deriv_nz - jac[1, 1]) < 10.**-6.
), 'jacobian d((lambda,nu))/d((R,z)) fails for (dn/dz)'
return None