def mplegenp(nu, mu, x):
if mu == int(mu) and x == 1:
# mpmath 0.17 gets this wrong
if mu == 0:
return 1
else:
return 0
return mpmath.legenp(nu, mu, x)
def mplegenp(nu, mu, x):
if mu == int(mu) and x == 1:
# mpmath 0.17 gets this wrong
if mu == 0:
return 1
else:
return 0
return mpmath.legenp(nu, mu, x)
def make_assoc_legendre_p_vals():
from mpmath import legenp
l = range(5)
m = range(-4, 5)
x = linspace('-0.99', '0.99', 67)
l, m, x = zip(*(vals for vals in outer(l, m, x) if abs(vals[1]) <= vals[0]))
p = [legenp(*vals, zeroprec = 1024) for vals in zip(l, m, x)]
return make_special_vals('assoc_legendre_p_vals', ('l', l), ('m', m), ('x', x), ('p', p))
def make_assoc_legendre_p_vals():
from mpmath import legenp
l = list(range(5))
m = list(range(-4, 5))
x = linspace('-0.99', '0.99', 67)
l, m, x = zip(*(vals for vals in outer(l, m, x) if abs(vals[1]) <= vals[0]))
p = [legenp(*vals, zeroprec = 1024) for vals in zip(l, m, x)]
return make_special_vals('assoc_legendre_p_vals', ('l', l), ('m', m), ('x', x), ('p', p))
def test_lpmv():
pts = []
for x in [-0.99, -0.557, 1e-6, 0.132, 1]:
pts.extend([
(1, 1, x),
(1, -1, x),
(-1, 1, x),
(-1, -2, x),
(1, 1.7, x),
(1, -1.7, x),
(-1, 1.7, x),
(-1, -2.7, x),
(1, 10, x),
(1, 11, x),
(3, 8, x),
(5, 11, x),
(-3, 8, x),
(-5, 11, x),
(3, -8, x),
(5, -11, x),
(-3, -8, x),
(-5, -11, x),
(3, 8.3, x),
(5, 11.3, x),
(-3, 8.3, x),
(-5, 11.3, x),
(3, -8.3, x),
(5, -11.3, x),
(-3, -8.3, x),
(-5, -11.3, x),
])
dataset = [p + (mpmath.legenp(p[1], p[0], p[2]), ) for p in pts]
dataset = np.array(dataset, dtype=np.float_)
evf = lambda mu, nu, x: sc.lpmv(mu.astype(int), nu, x)
olderr = np.seterr(invalid='ignore')
try:
FuncData(evf, dataset, (0, 1, 2), 3, rtol=1e-10, atol=1e-14).check()
finally:
np.seterr(**olderr)
def test_lpmv():
pts = []
for x in [-0.99, -0.557, 1e-6, 0.132, 1]:
pts.extend([
(1, 1, x),
(1, -1, x),
(-1, 1, x),
(-1, -2, x),
(1, 1.7, x),
(1, -1.7, x),
(-1, 1.7, x),
(-1, -2.7, x),
(1, 10, x),
(1, 11, x),
(3, 8, x),
(5, 11, x),
(-3, 8, x),
(-5, 11, x),
(3, -8, x),
(5, -11, x),
(-3, -8, x),
(-5, -11, x),
(3, 8.3, x),
(5, 11.3, x),
(-3, 8.3, x),
(-5, 11.3, x),
(3, -8.3, x),
(5, -11.3, x),
(-3, -8.3, x),
(-5, -11.3, x),
])
dataset = [p + (mpmath.legenp(p[1], p[0], p[2]),) for p in pts]
dataset = np.array(dataset, dtype=np.float_)
evf = lambda mu,nu,x: sc.lpmv(mu.astype(int), nu, x)
olderr = np.seterr(invalid='ignore')
try:
FuncData(evf, dataset, (0,1,2), 3, rtol=1e-10, atol=1e-14).check()
finally:
np.seterr(**olderr)
def eqn42_solve( geom, Nmax ):
# eqn 4.2
NMAX = Nmax # where to truncate the infinite system of eqns
S = geom['S']
distanceqp = geom['distanceqp']
thetaqp = geom['thetaqp']
phiqp = geom['phiqp']
radii = geom['radii']
# coefficient matrix
if USEMPMATH:
CM = mpmath.zeros( S*NMAX*(2*NMAX+1) )
else:
CM = np.zeros( (S*NMAX*(2*NMAX+1), S*NMAX*(2*NMAX+1)) )
for sprimei,sprime in enumerate(range(S)):
for nprimei,nprime in enumerate(range(NMAX)):
for mprimei,mprime in enumerate(range(-nprime,nprime+1)):
# row index
ri = sprimei*NMAX*(2*NMAX+1) + nprimei*(2*NMAX+1) + mprimei
# row prefactors
prefac = (-1)**(nprime+mprime) * radii[sprimei]**(nprime+1) \
* 1./factorial(nprime+mprime)
for si,s in enumerate(range(S)):
if sprimei != si:
for ni,n in enumerate(range(NMAX)):
for mi,m in enumerate(range(-n,n+1)):
# column index
ci = si*NMAX*(2*NMAX+1) + ni*(2*NMAX+1) + mi
f1 = distanceqp[sprimei,si]**(-(n+1))
f2 = (radii[sprimei]/distanceqp[sprimei,si])**nprime
f3 = factorial(n-m+nprime+mprime)/factorial(n-m)
if USEMPMATH:
f4 = mpmath.legenp( n+nprime, m-mprime, \
np.cos(thetaqp[sprimei,si]) )
else:
f4 = scipy.special.lpmv( m-mprime, n+nprime, \
np.cos(thetaqp[sprimei,si]) )
f5 = np.exp( 1j*(m-mprime)*phiqp[sprimei,si] )
CM[ri,ci] = prefac*f1*f2*f3*f4*f5
if USEMPMATH:
CM += mpmath.diag(np.ones(S*NMAX*(2*NMAX+1)))
Qs = mpmath.zeros(S)
else:
CM += np.diag(np.ones(S*NMAX*(2*NMAX+1)))
Qs = np.zeros((S,S))
for si in range(S):
if USEMPMATH:
rhs = mpmath.zeros( S*NMAX*(2*NMAX+1), 1 )
else:
rhs = np.zeros((CM.shape[0],))
rhs[si*NMAX*(2*NMAX+1):(si*NMAX+1)*(2*NMAX+1)] = radii[si]
if USEMPMATH:
sol = mpmath.lu_solve( CM, rhs )
else:
sol = np.linalg.solve( CM, rhs )
#print sol[::(NMAX*(2*NMAX+1))]
Qs[si,:] = sol[::(NMAX*(2*NMAX+1))]
return Qs
