qpms/tests/transcoeff_cruzan.py

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# [Xu] = Journal of computational physics 139, 137165
from __future__ import print_function
def p_q(q, n, nu):
return n + nu - 2*q
def qmax(M, n, mu, nu):
return floor(min(n, nu, (n+nu-abs(M+mu))/2))
def Qmax(M, n, mu, nu): # [Xu](60)
return floor(min(n, nu, (n+nu+1-abs(M+mu))/2))
def gaunta_p(M, n, mu, nu, p): # [Xu](5)
#print (M,n,mu,nu,p, file=sys.stderr)
return (-1)**(M+mu) * (2*p +1) * sqrt(
factorial(n+M) * factorial(nu+mu) * factorial(p-M-mu)
/ factorial(n-M) / factorial(nu-mu) / factorial(p+M+mu)) * (
wigner_3j(n, nu, p, 0, 0, 0) * wigner_3j(n, nu, p, M, mu, -M-mu))
def bCXcoeff(M, n, mu, nu, p): # [Xu](61)
#print(M,n,mu,nu,p,file=sys.stderr)
return (-1)**(M+mu) * (2*p + 3) * sqrt(
factorial(n+M) * factorial(nu+mu) * factorial(p+1-M-mu)
/ factorial(n-M) / factorial(nu-mu) / factorial(p+1+M+mu)) * (
wigner_3j(n, nu, p, 0, 0, 0) * wigner_3j(n, nu, p+1, M, mu, -M-mu))
def ACXcoeff(m, n, mu, nu, q): # [Xu](58)
p = p_q(q,n,nu)
return ((-1)**m * (2*nu + 1) * factorial(n+m) * factorial(nu-mu) / (
2 * n * (n+1) * factorial(n-m) * factorial(nu+mu)) * I**p *
(n*(n+1) + nu*(nu+1) - p*(p+1)) * gaunta_p(-m,n,mu,nu,p))
def BCXcoeff(m, n, mu, nu, q): # [Xu](59)
p = p_q(q,n,nu)
return ((-1)**(m+1) * (2*nu + 1) * factorial(n+m) * factorial(nu-mu) / (
2 * n * (n+1) * factorial(n-m) * factorial(nu+mu)) * I**(p+1) *
sqrt(((p+1)**2-(n-nu)**2) * ((n+nu+1)**2-(p+1)**2))
* bCXcoeff(-m,n,mu,nu,p))
def printACXcoeffs(lMax, file=sys.stdout):
for n in IntegerRange(lMax+1):
for nu in IntegerRange(lMax+1):
for m in IntegerRange(-n, n+1):
for mu in IntegerRange(-nu, nu+1):
for q in IntegerRange(qmax(-m,n,mu,nu)):
#print(m, n, mu, nu, q, p_q(q,n,nu), file=sys.stderr)
coeff= ACXcoeff(m, n, mu, nu, q);
print(N(coeff, prec=53),
", // %d, %d, %d, %d, %d," % (m,n,mu,nu,q),
coeff,
file=file)
return
def printBCXcoeffs(lMax, file=sys.stdout):
for n in IntegerRange(lMax+1):
for nu in IntegerRange(lMax+1):
for m in IntegerRange(-n, n+1):
for mu in IntegerRange(-nu, nu+1):
for q in IntegerRange(1, Qmax(-m,n,mu,nu) +1 ):
#print(m, n, mu, nu, q, p_q(q,n,nu), file=sys.stderr)
coeff= BCXcoeff(m, n, mu, nu, q);
print(N(coeff, prec=53),
", // %d, %d, %d, %d, %d," % (m,n,mu,nu,q),
coeff,
file=file)
return
sphericalBessels = (None,
spherical_bessel_J,
spherical_bessel_Y,
spherical_hankel1,
spherical_hankel2
)
# N.B. sage's gen_legendre_P _does_ include (-1)**m Condon-Shortley phase
# whereas formulae in [Xu] do not.
def trcoeff_ACX(m, n, mu, nu, besseltype, kd, th, fi, csphase=1): # [Xu](58)
res = 0
for q in range(qmax(-m,n,mu,nu)+1):
p = p_q(q,n,nu)
res += ACXcoeff(m,n,mu,nu,q) * sphericalBessels[besseltype](p,kd) * gen_legendre_P(p, mu-m, cos(th)) * (-csphase)**(mu-m) # compensate for csphase
res *= exp(I*(mu-m)*fi)
return res
def trcoeff_BCX(m, n, mu, nu, besseltype, kd, th, fi, csphase=1): # [Xu](59)
res = 0
for q in IntegerRange(1,Qmax(-m,n,mu,nu)+1):
p = p_q(q,n,nu)
res += BCXcoeff(m,n,mu,nu,q) * sphericalBessels[besseltype](p+1,kd) * gen_legendre_P(p+1, mu-m, cos(th)) * (-csphase)**(mu-m)
res *= exp(I*(mu-m)*fi)
return res
def legpi_xu(n, m, fi): # momentálně neošetřeny okraje (cos(fi) == +- 1)
return m/sin(fi) * gen_legendre_P(n, m, cos(fi))
def legtau_xu(n, m, fi):
locx = var('locx')
return -sin(fi)*derivative(gen_legendre_P(n,m,locx), locx).substitute(locx = cos(fi))
def vswf_M_xu(besseltype, n, m, kr, th, fi):
postpart = sphericalBessels[besseltype](n, kr) * exp(I * m * fi)
tc = I*legpi_xu(n,m,th) * postpart
fc = -legtau_xu(n,m,th) * postpart
return (0, tc, fc)
def vswf_N_xu(besseltype, n, m, kr, th, fi):
eimf = exp(I * m * fi)
rc = n * (n+1) * gen_legendre_P(n, m, cos(th)) * sphericalBessels[besseltype](n, kr)/kr * eimf
krv = var('krv')
radpart = derivative(krv * sphericalBessels[besseltype](n, krv), krv).substitute(krv=kr)/kr
tc = legtau_xu(n,m,th) * radpart * eimf
fc = I*legpi_xu(n,m,th) * radpart * eimf
return (rc,tc,fc)
def cart2sph(v):
(x, y, z) = v
r = sqrt(x**2 + y**2 + z**2)
th = arccos(z/r) if r else 0
fi = arctan2(y,x)
return (r, th, fi)
def sph2cart(s):
(r, th, fi) = s
sinth = sin(th)
x = r * sinth * cos(fi)
y = r * sinth * sin(fi)
z = r * cos(th)
return (x,y,z)
def sphvec2cart(loccart, sph):
r, th, fi = sph
sinth = sin(th)
costh = cos(th)
sinfi = sin(fi)
cosfi = cos(fi)
rx = sinth * cosfi
ry = sinth * sinfi
rz = costh
tx = costh * cosfi
ty = costh * sinfi
tz = -sinth
fx = -sinfi
fy = cosfi
fz = 0
rc, tc, fc = loccart
x = rx * rc + tx * tc + fx * fc
y = ry * rc + ty * tc + fy * fc
z = rz * rc + tz * tc + fz * fc
return (x, y, z)
def cart2sphvec(cart, sph):
_, th, fi = sph
x, y, z = cart
rx = sinth * cosfi
ry = sinth * sinfi
rz = costh
tx = costh * cosfi
ty = costh * sinfi
tz = -sinth
fx = -sinfi
fy = cosfi
fz = 0
rc = rx * x + ry * y + rz * z
tc = tx * x + ty * y + tz * z
fc = fx * x + fy * y + fz * z
return (rc, tc, fc)
def test_M_translation_xu(lMax, origl, origm, origcartat, cartshift):
ox, oy, oz = origcartat
sx, sy, sz = cartshift
newcartat = (ox - sx, oy - sy, oz - sz)
w1s = cart2sph(origcartat)
w2s = cart2sph(newcartat)
pass # TODO