def operator(op, N, k, ell, a=alpha):
q, m = k + a, ell + 1 / 2
# zeros
if (op == '0'): return jacobi.operator('0', N, q, m)
# identity
if (op == 'I'): return jacobi.operator('I', N, q, m)
# conversion
if (op == 'E'): return jacobi.operator('A+', N, q, m, rescale=np.sqrt(0.5))
# derivatives
if (op == 'D-'): return jacobi.operator('C+', N, q, m, rescale=2.0)
if (op == 'D+'): return jacobi.operator('D+', N, q, m, rescale=2.0)
# r multiplication
if (op == 'R-'):
return jacobi.operator('B-', N, q, m, rescale=np.sqrt(0.5))
if (op == 'R+'):
return jacobi.operator('B+', N, q, m, rescale=np.sqrt(0.5))
# z = 2*r*r-1 multiplication
if op == 'Z': return jacobi.operator('J', N, q, m)
if op == 'r=1':
return jacobi.operator('z=+1', N, q, m, rescale=np.sqrt(2.0))
def operator(op,N_max,k,m,a=alpha):
q = k + a
N = N_max # - some function of m
# null
if (op == '0'): return jacobi.operator('0',N,q,m)
# identity
if (op == 'I'): return jacobi.operator('I',N,q,m)
# conversion
if (op == 'E'): return jacobi.operator('A+',N,q,m,rescale=np.sqrt(0.5))
# derivatives
if (op == 'D-'): return jacobi.operator('C+',N,q,m,rescale=np.sqrt(2.0))
if (op == 'D+'): return jacobi.operator('D+',N,q,m,rescale=np.sqrt(2.0))
# r multiplication
if (op == 'R-'): return jacobi.operator('B-',N,q,m,rescale=np.sqrt(0.5))
if (op == 'R+'): return jacobi.operator('B+',N,q,m,rescale=np.sqrt(0.5))
# z = 2*r*r-1 multiplication
if op == 'Z': return jacobi.operator('J',N,q,m)
if op == 'r=1': return jacobi.operator('z=+1',N,q,m,rescale=np.sqrt(0.5))
def jacobi_matrix(N, a, b):
J = jacobi.operator('J', N - 1, a, b)
return J.tocsr().astype(np.float64)
def operator(op,L_max,m,s):
""" Various derivative and multiplication operators for spin-weighted spherical harmonics .
Parameters
----------
L_max, m, s: int
spherical harmonic parameters
op: string = 'I', 'C', 'k+', 'k-', 'S+', 'S-'
I = Identity
k+ = sqrt(0.5)*(Grad_theta + i Grad_phi) with (m,s) -> (m,s+1); diagonal in ell
k- = sqrt(0.5)*(Grad_theta - i Grad_phi) with (m,s) -> (m,s-1); diagonal in ell
C = Cosine multiplication with (m,s) -> (m,s); couples ell
S+ = Sine multiplication with (m,s) -> (m,s+1); couples ell
S- = Sine multiplication with (m,s) -> (m,s-1); couples ell
"""
def ds(op):
"""get s increment from the operator string"""
if len(op)==1: return 0
return int(op[1]+'1')
mu, N = ds(op), L_max-L_min(m,s)
a,b,da,db = a_and_b(m,s,ds=mu)
rescale = -mu*np.sqrt(0.5)
# identity
if op == 'I':
a,b = a_and_b(m,s)
return jacobi.operator('I',N,a,b)
# cosine multiplication
if op == 'C':
a,b = a_and_b(m,s)
return jacobi.operator('J',N,a,b)
# derivatives
if (op == 'k+') or (op=='k-'):
if (da== 1) and (db==-1): return jacobi.operator('C+',N ,a,b,rescale=rescale)
if (da==-1) and (db== 1): return jacobi.operator('C-',N ,a,b,rescale=rescale)
if (da== 1) and (db== 1): return jacobi.operator('D+',N ,a,b,rescale=rescale)[:-1,:]
if (da==-1) and (db==-1): return jacobi.operator('D-',N+1,a,b,rescale=rescale)[:,:-1]
# sine multiplication
if (op == 'S+') or (op=='S-'):
if (da== 1) and (db==-1):
A = jacobi.operator('A+',N+1,a, b)
B = jacobi.operator('B-',N+1,a+1,b)
return B.dot(A)[:-1,:-1]
if (da==-1) and (db== 1):
A = jacobi.operator('A-',N+1,a, b)
B = jacobi.operator('B+',N+1,a-1,b)
return B.dot(A)[:-1,:-1]
if (da== 1) and (db== 1):
A = jacobi.operator('A+',N+1,a, b)
B = jacobi.operator('B+',N+1,a+1,b)
return (B.dot(A))[:-2,:-1]
if (da==-1) and (db==-1):
A = jacobi.operator('A-',N+2,a, b)
B = jacobi.operator('B-',N+2,a-1,b)
return (B.dot(A))[:-1,:-2]