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spin.py
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executable file
·298 lines (212 loc) · 7.4 KB
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#! /bin/python
import numpy as np
from scipy.special import loggamma as lgm, logsumexp
from alp_integral import logI2
from scipy.special import assoc_legendre_p
#import matplotlib.pyplot as plt
#from matplotlib.transforms import Affine2D
#import spherical
#import quaternionic
def plot_sphere(theta, phi, img, title = None, colorbar = True):
fig = plt.figure()
ax = fig.add_subplot(111, projection='mollweide')
cax = fig.add_axes([0.27, 0.8, 0.5, 0.05])
x = phi - np.pi # Wrap to [-π, π]
y = np.pi/2 - theta # Latitude ([-π/2, π/2])
pcm = ax.pcolormesh(x, y, img,
cmap = "hot",
)
if colorbar:
fig.colorbar(pcm, cax=cax, orientation='horizontal')
if title is not None:
cax.set_title(title)
return fig, ax
def E1p(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-2, m1+2, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
log_coef = np.log(abs((l1 - m1**2) / (4*m1) / (m1+1)))
sign_coef = np.sign((l1 - m1**2) / (4*m1) / (m1+1))
return logI + log_coef, sign_I * sign_coef
def E2p(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-2, m1, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
log_coef = np.log(abs((l1 - m1**2) * (l1 + m1 - 1) * (l1 + m1)/ 2.0 / (m1**2 - 1)))
sign_coef = np.sign((l1 - m1**2) * (l1 + m1 - 1) * (l1 + m1)/ 2.0 / (m1**2 - 1))
return logI + log_coef, sign_I * sign_coef
def E3p(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-2, m1-2, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
term1 = (l1 + m1) * (l1 + m1 - 1) * (l1 + m1 - 2) * (l1 + m1 - 3)
log_coef = np.log(abs(term1 * (l1 - m1**2) / (4*m1) / (m1-1)))
sign_coef = np.sign(term1 * (l1 - m1**2) / (4*m1) / (m1-1))
return logI + log_coef, sign_I * sign_coef
def E4p(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1, m1, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
log_coef = np.log(abs(.5 * l1 * (l1 - 1)))
sign_coef = np.sign(.5 * l1 * (l1 - 1))
return logI + log_coef, sign_I * sign_coef
def E5p(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-2, m1+2, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
log_coef = np.log(abs(-(l1 + m1) / (4*m1) / (m1 + 1)))
sign_coef = np.sign(-(l1 + m1) / (4*m1) / (m1 + 1))
return logI + log_coef, sign_I * sign_coef
def E6p(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-2, m1, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
log_coef = np.log(abs(-(l1+m1) * (l1-1) * (l1 + m1 - 1) / 2.0 / (m1**2 - 1)))
sign_coef = np.sign(-(l1+m1) * (l1-1) * (l1 + m1 - 1) / 2.0 / (m1**2 - 1))
return logI + log_coef, sign_I * sign_coef
def E7p(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-2, m1-2, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
term1 = (l1 + m1 - 1) * (l1 + m1 - 2) * (l1 + m1 - 3)
log_coef = np.log(abs(-term1 * (l1+m1) * (l1-m1) / (4*m1) /(m1-1)))
sign_coef = np.sign(-term1 * (l1+m1) * (l1-m1) / (4*m1) /(m1-1))
return logI + log_coef, sign_I * sign_coef
def E1m(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-1, m1+2, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
log_coef = np.log(abs((l1 - 1) / (4 * (m1 + 1))))
sign_coef = np.sign((l1 - 1) / (4 * (m1 + 1)))
return logI + log_coef, sign_I * sign_coef
def E2m(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-1, m1, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
log_coef = np.log(abs((l1 - 1)/2 * l1 * m1 * (l1 + m1) / (m1**2 - 1)))
sign_coef = np.sign((l1 - 1)/2 * l1 * m1 * (l1 + m1) / (m1**2 - 1))
return logI + log_coef, sign_I * sign_coef
def E3m(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-1, m1-2, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
term1 = (l1 + m1) * (l1 + m1 - 1) * (l1 + m1 - 2)
term2 = (l1 - m1 + 1)
log_coef = np.log(abs((l1 - 1) * term1 * term2 / (4 * (m1 - 1))))
sign_coef = np.sign((l1 - 1) * term1 * term2 / (4 * (m1 - 1)))
return logI + log_coef, sign_I * sign_coef
def E4m(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-3, m1+2, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
log_coef = np.log(abs(-(l1 + m1) / (4 * (m1 + 1))))
sign_coef = np.sign(-(l1 + m1) / (4 * (m1 + 1)))
return logI + log_coef, sign_I * sign_coef
def E5m(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-3, m1, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
log_coef = np.log(abs(-(l1+m1)/2 * m1 * (l1+m1-2) * (l1+m1-1) / (m1**2 - 1)))
sign_coef = np.sign(-(l1+m1)/2 * m1 * (l1+m1-2) * (l1+m1-1) / (m1**2 - 1))
return logI + log_coef, sign_I * sign_coef
def E6m(l, m, l1):
m1 = -m
logI, sign_I = logI2(l1-3, m1-2, l, m)
if sign_I == 0.0:
return -np.inf, 0.0
term1 = (l1 + m1 - 1) * (l1 + m1 - 2) * (l1 + m1 - 3) * (l1 + m1 - 4)
log_coef = np.log(abs(-(l1+m1)/4 * term1 / (m1 - 1)))
sign_coef = np.sign(-(l1+m1)/4 * term1 / (m1 - 1))
return logI + log_coef, sign_I * sign_coef
def intY2Y(l, m, l1):
m1 = -m
logN_l1 = .5 * (lgm(l1-1) - lgm(l1+3))
logK_l1 = .5 * np.log((2*l1 + 1)/(4*np.pi)) + \
.5 * (lgm(l1 - m1 + 1) - lgm(l1 + m1 +1))
logK_l = .5 * np.log((2*l + 1)/(4*np.pi)) + \
.5 * (lgm(l - m + 1) - lgm(l + m +1))
log_C = np.log(2) + logN_l1 + logK_l1 + logK_l
sign_C = -1.0
#coef2 = (-1)**m * np.exp(lgm(l+m+1) - lgm(l-m+1)) / (2*np.pi)
E_terms = np.array([
E1p(l, m, l1),
E2p(l, m, l1),
E3p(l, m, l1),
E4p(l, m, l1),
E5p(l, m, l1),
E6p(l, m, l1),
E7p(l, m, l1),
E1m(l, m, l1),
E2m(l, m, l1),
E3m(l, m, l1),
E4m(l, m, l1),
E5m(l, m, l1),
E6m(l, m, l1)
])
E_terms = E_terms[E_terms[:, 1] != 0.0]
logE = E_terms[:, 0]
signs = E_terms[:, 1]
if len(logE) == 0:
return 0.0
summ, sign = logsumexp(logE, b=signs, return_sign=True)
summ = summ + log_C
sign = sign * sign_C
return sign * np.exp(summ)
def F(l):
return 4*np.pi * (-1)**l / np.sqrt((l + 2)*(l + 1)*l*(l - 1))
def N(l):
logN_l = .5 * (lgm(l-1) - lgm(l+3))
return np.exp(logN_l)
def K(l, m):
logK = .5 * np.log((2*l + 1)/(4*np.pi)) + \
.5 * (lgm(l - m + 1) - lgm(l + m +1))
return np.exp(logK)
def Y(l, m, theta, phi):
leg = assoc_legendre_p(m, n, np.cos(theta))
k = K(l, m)
return k * leg * np.exp(1j*m * phi)
def p_inv_factor(l, m):
log_fac = lgm(l - m + 1) - lgm(l + m + 1)
return (-1)**m * np.exp(log_fac)
def E(l, m):
for l1 in range(np.abs(m), np.abs(m) + 5):
E_terms = np.array([
E1p(l, m, l1),
E2p(l, m, l1),
E3p(l, m, l1),
E4p(l, m, l1),
E5p(l, m, l1),
E6p(l, m, l1),
E7p(l, m, l1),
E1m(l, m, l1),
E2m(l, m, l1),
E3m(l, m, l1),
E4m(l, m, l1),
E5m(l, m, l1),
E6m(l, m, l1)
])
if __name__ == "__main__":
#E(3, 3)
# fig, ax = plot_sphere(theta, phi, Y2[:, :, idx].real, title = f"${{}}_2 Y_{{{l},{m}}}$")
# fig, ax = plot_sphere(theta, phi, Y[:, :, idx].real, title = f"$Y_{{{l},{m}}}$")
#
#
# fig, ax = plot_sphere(theta, phi, (Y2[:, :, idx] * Y[:, :, idx]).real,
# title = f"${{}}_2 Y_{{{l},{m}}} \cdot Y_{{{l},{m}}}$")
#
#
#
#
# save_image("123.pdf")