def get_omegakappatau(n, m):
"""Return omega, kappa, tau integrals, required for the scattering
cross-section evaluation"""
dim = n - m + 1
delta11 = identity(dim)
if dim - 1 > 0:
delta21 = diag(ones(dim - 1), -1)
delta12 = diag(ones(dim - 1), 1)
else:
delta21 = 0
delta12 = 0
if dim - 2 > 0:
delta31 = diag(ones(dim - 2), -2)
delta13 = diag(ones(dim - 2), 2)
else:
delta31 = 0
delta13 = 0
l = arange(m, n + 1)
ff = factorial(l + m) / factorial(l - m) * 2 / (2 * l + 1.)
ff = transpose(repeat([ff], len(l), 0))
l = transpose(repeat([l], len(l), 0))
omega = ( 2 * (l * (l + 1) + m ** 2 - 1) / ((2 * l + 3) * (2 * l - 1.)) * delta11\
- (l - m) * (l - m - 1) / ((2 * l - 1) * (2 * l - 3.)) * delta31\
- (l + m + 1) * (l + m + 2) / ((2 * l + 3) * (2 * l + 5.)) * delta13 )\
* ff
kappa = ( (l + 1) * (l - m) / (2 * l - 1.) * delta21\
- l * (l + m + 1) / (2 * l + 3.) * delta12 )\
* ff
tau = l * (l + 1) * ff * delta11
return omega, kappa, transpose(kappa), tau
def get_omegakappatau(n, m):
"""Return omega, kappa, tau integrals, required for the scattering
cross-section evaluation"""
dim = n - m + 1
delta11 = identity(dim)
if dim - 1 > 0:
delta21 = diag(ones(dim - 1), -1)
delta12 = diag(ones(dim - 1), 1)
else:
delta21 = 0
delta12 = 0
if dim - 2 > 0:
delta31 = diag(ones(dim - 2), -2)
delta13 = diag(ones(dim - 2), 2)
else:
delta31 = 0
delta13 = 0
l = arange(m, n + 1)
ff = factorial(l + m) / factorial(l - m) * 2 / (2 * l + 1.)
ff = transpose(repeat([ff], len(l), 0))
l = transpose(repeat([l], len(l), 0))
omega = ( 2 * (l * (l + 1) + m ** 2 - 1) / ((2 * l + 3) * (2 * l - 1.)) * delta11\
- (l - m) * (l - m - 1) / ((2 * l - 1) * (2 * l - 3.)) * delta31\
- (l + m + 1) * (l + m + 2) / ((2 * l + 3) * (2 * l + 5.)) * delta13 )\
* ff
kappa = ( (l + 1) * (l - m) / (2 * l - 1.) * delta21\
- l * (l + m + 1) / (2 * l + 3.) * delta12 )\
* ff
tau = l * (l + 1) * ff * delta11
return omega, kappa, transpose(kappa), tau
def legendre_(l, m, x):
"""
Legendre polynomial.
Check equation (3) from Townsend, 2002:
>>> ls,x = [0,1,2,3,4,5],cos(linspace(0,pi,100))
>>> check = 0
>>> for l in ls:
... for m in range(-l,l+1,1):
... Ppos = legendre_(l,m,x)
... Pneg = legendre_(l,-m,x)
... mycheck = Pneg,(-1)**m * factorial(l-m)/factorial(l+m) * Ppos
... check += sum(abs(mycheck[0]-mycheck[1])>1e-10)
>>> print check
0
"""
m_ = abs(m)
legendre_poly = legendre(l)
deriv_legpoly_ = legendre_poly.deriv(m=m_)
deriv_legpoly = np.polyval(deriv_legpoly_, x)
P_l_m = (-1)**m_ * (1 - x**2)**(m_ / 2.) * deriv_legpoly
if m < 0:
P_l_m = (-1)**m_ * factorial(l - m_) / factorial(l + m_) * P_l_m
return P_l_m
def legendre_(l,m,x):
"""
Legendre polynomial.
Check equation (3) from Townsend, 2002:
>>> ls,x = [0,1,2,3,4,5],cos(linspace(0,pi,100))
>>> check = 0
>>> for l in ls:
... for m in range(-l,l+1,1):
... Ppos = legendre_(l,m,x)
... Pneg = legendre_(l,-m,x)
... mycheck = Pneg,(-1)**m * factorial(l-m)/factorial(l+m) * Ppos
... check += sum(abs(mycheck[0]-mycheck[1])>1e-10)
>>> print check
0
"""
m_ = abs(m)
legendre_poly = legendre(l)
deriv_legpoly_ = legendre_poly.deriv(m=m_)
deriv_legpoly = np.polyval(deriv_legpoly_,x)
P_l_m = (-1)**m_ * (1-x**2)**(m_/2.) * deriv_legpoly
if m<0:
P_l_m = (-1)**m_ * factorial(l-m_)/factorial(l+m_) * P_l_m
return P_l_m
def splitter(self, key, line):
"""The mapper for step 1. Splits the range of possible tours into
reasonably sized chunks for the consumption of the step 2 mappers.
At this point the 'line' input should come directly from the first line
of the one-line json file contains the edge cost graph and the starting
node. The key is not relevant.
"""
#loading the json description of the trip to get at the size
#of the edge costgraph
sales_trip = json.loads(line)
m = numpy.matrix(sales_trip['graph'])
num_nodes = m.shape[0]
num_tours = factorial(num_nodes - 1)
#Here we break down the full range of possible tours into smaller
#pieces. Each piece is passed along as a key along with the trip
#description.
step_size = int(100 if num_tours < 100**2 else num_tours / 100)
steps = range(0, num_tours, step_size) + [num_tours]
ranges = zip(steps[0:-1], steps[1:])
for range_low, range_high in ranges:
#The key prepresents the range of tours to cost
yield (("%d-%d" % (range_low, range_high), sales_trip))
def splitter(self, key, line):
"""The mapper for step 1. Splits the range of possible tours into
reasonably sized chunks for the consumption of the step 2 mappers.
At this point the 'line' input should come directly from the first line
of the one-line json file contains the edge cost graph and the starting
node. The key is not relevant.
"""
#loading the json description of the trip to get at the size
#of the edge costgraph
sales_trip = json.loads(line)
m = numpy.matrix(sales_trip['graph'])
num_nodes = m.shape[0]
num_tours = factorial(num_nodes - 1)
#Here we break down the full range of possible tours into smaller
#pieces. Each piece is passed along as a key along with the trip
#description.
step_size = int(100 if num_tours < 100**2 else num_tours / 100)
steps = range(0, num_tours, step_size) + [num_tours]
ranges = zip(steps[0:-1], steps[1:])
for range_low, range_high in ranges:
#The key prepresents the range of tours to cost
yield( ("%d-%d"%(range_low,range_high), sales_trip ))
def WignerD( the , l , m , n ) :
"""
Wigner D matrices: d_{mn}^{l}(the)
the -- angle in radians
l -- degree (integer)
m -- order (integer)
n -- order (integer)
"""
if m >= n :
factor = (-1)**(l-m) * np.sqrt( ( scimcm.factorial( l+m ) * scimcm.factorial( l-m ) ) /
( scimcm.factorial( l+n ) * scimcm.factorial( l-n ) ) )
a = m + n
b = m - n
print a
print b
return factor * np.cos( the/2. )**a * ( -np.sin( the/2. ) )**b * scisp.jacobi( l-m , a , b )( -np.cos( the ) )
else :
return (-1)**(m-n) * WignerD( the , l , n , m )
def factorial(n, method='reduce'):
"""
Compute the factorial n! using long integers.
Different implementations are available
(see source code for the methods).
Here is an efficiency comparison of the methods (computing 80!):
reduce | 1.00
lambda list comprehension | 1.70
lambda functional | 3.08
plain recursive | 5.83
lambda recursive | 21.73
scipy | 131.18
"""
if not isinstance(n, (int, long, float)):
raise TypeError, 'factorial(n): n must be integer not %s' % type(n)
n = long(n)
if n == 0 or n == 1:
return 1
if method == 'plain iterative':
f = 1
for i in range(1, n+1):
f *= i
return f
elif method == 'plain recursive':
if n == 1:
return 1
else:
return n*factorial(n-1, method)
elif method == 'lambda recursive':
fc = lambda n: n and fc(n-1)*long(n) or 1
return fc(n)
elif method == 'lambda functional':
fc = lambda n: n<=0 or \
reduce(lambda a,b: long(a)*long(b), xrange(1,n+1))
return fc(n)
elif method == 'lambda list comprehension':
fc = lambda n: [j for j in [1] for i in range(2,n+1) \
for j in [j*i]] [-1]
return fc(n)
elif method == 'reduce':
return reduce(operator.mul, xrange(2, n+1))
elif method == 'scipy':
try:
import scipy.misc.common as sc
return sc.factorial(n)
except ImportError:
print 'numpyutils.factorial: scipy is not available'
print 'default method="reduce" is used instead'
return reduce(operator.mul, xrange(2, n+1))
# or return factorial(n)
else:
raise ValueError, 'factorial: method="%s" is not supported' % method
def factorial(n, method='reduce'):
"""
Compute the factorial n! using long integers.
Different implementations are available
(see source code for the methods).
Here is an efficiency comparison of the methods (computing 80!):
reduce | 1.00
lambda list comprehension | 1.70
lambda functional | 3.08
plain recursive | 5.83
lambda recursive | 21.73
scipy | 131.18
"""
if not isinstance(n, (int, long, float)):
raise TypeError, 'factorial(n): n must be integer not %s' % type(n)
n = long(n)
if n == 0 or n == 1:
return 1
if method == 'plain iterative':
f = 1
for i in range(1, n + 1):
f *= i
return f
elif method == 'plain recursive':
if n == 1:
return 1
else:
return n * factorial(n - 1, method)
elif method == 'lambda recursive':
fc = lambda n: n and fc(n - 1) * long(n) or 1
return fc(n)
elif method == 'lambda functional':
fc = lambda n: n<=0 or \
reduce(lambda a,b: long(a)*long(b), xrange(1,n+1))
return fc(n)
elif method == 'lambda list comprehension':
fc = lambda n: [j for j in [1] for i in range(2,n+1) \
for j in [j*i]] [-1]
return fc(n)
elif method == 'reduce':
return reduce(operator.mul, xrange(2, n + 1))
elif method == 'scipy':
try:
import scipy.misc.common as sc
return sc.factorial(n)
except ImportError:
print 'numpyutils.factorial: scipy is not available'
print 'default method="reduce" is used instead'
return reduce(operator.mul, xrange(2, n + 1))
# or return factorial(n)
else:
raise ValueError, 'factorial: method="%s" is not supported' % method
def cmdet(d):
# compute the CMD determinant of the euclidean distance matrix d
# -> d should not be squared!
D = np.ones((d.shape[0] + 1, d.shape[0] + 1))
D[0, 0] = 0.0
D[1:, 1:] = d**2
j = np.float32(D.shape[0] - 2)
f1 = (-1.0)**(j + 1) / ((2**j) * ((factorial(j))**2))
cmd = f1 * np.linalg.det(D)
# sometimes, for very small values "cmd" might be negative ...
return np.sqrt(np.abs(cmd))
def cmdet(d): # compute the CMD determinant of the euclidean distance matrix d # -> d should not be squared! D = np.ones((d.shape[0]+1,d.shape[0]+1)) D[0,0] = 0.0 D[1:,1:] = d**2 j = np.float32(D.shape[0]-2) f1 = (-1.0)**(j+1) / ( (2**j) * ((factorial(j))**2)) cmd = f1 * np.linalg.det(D) # sometimes, for very small values "cmd" might be negative ... return np.sqrt(np.abs(cmd))
def get_inc(self, n, m, axisymm=False):
"""Return incident field expansion coefficients"""
#if (self.Pna is None) or (len(self.Pna)!=n):
self.Pna, self.Pdna = lpmn(n, n, cos(self.alpha))[:]
self.sina = sin(self.alpha)
sina = self.sina
Pna = self.Pna[m, m:]
l = arange(m, n + 1)
if axisymm:
c_inc = 1j**l * (2 * l + 1.) / (l * (l + 1.)) * Pna
else:
Pdna = self.Pdna[m, m:]
c_inc = zeros(2 * (n - m + 1), dtype=complex)
if sina == 0. and m == 1:
c_inc[:n - m + 1] = -1j**(l - 1) * (2 * l + 1)
else:
c_inc[:n - m + 1] = -1j ** (l - 1) / sina\
* 2 * (2 * l + 1)\
* factorial(l - m) / factorial(l + m)\
* Pna
return c_inc
def get_inc(self, n, m, axisymm=False):
"""Return incident field expansion coefficients"""
#if (self.Pna is None) or (len(self.Pna)!=n):
self.Pna, self.Pdna = lpmn(n, n, cos(self.alpha))[:]
self.sina = sin(self.alpha)
sina = self.sina
Pna = self.Pna[m, m:]
l = arange(m, n + 1)
if axisymm:
c_inc = 1j ** l * (2 * l + 1.) / (l * (l + 1.)) * Pna
else:
Pdna = self.Pdna[m, m:]
c_inc = zeros(2 * (n - m + 1), dtype=complex)
if sina == 0. and m == 1:
c_inc[:n - m + 1] = -1j ** (l - 1) * (2 * l + 1)
else:
c_inc[:n - m + 1] = -1j ** (l - 1) / sina\
* 2 * (2 * l + 1)\
* factorial(l - m) / factorial(l + m)\
* Pna
return c_inc
def sph_harm(theta,phi,l=2,m=1):
"""
Spherical harmonic according to Townsend, 2002.
This function is memoized: once a spherical harmonic is computed, the
result is stored in memory
>>> theta,phi = mgrid[0:pi:20j,0:2*pi:40j]
>>> Ylm20 = sph_harm(theta,phi,2,0)
>>> Ylm21 = sph_harm(theta,phi,2,1)
>>> Ylm22 = sph_harm(theta,phi,2,2)
>>> Ylm2_2 = sph_harm(theta,phi,2,-2)
>>> p = figure()
>>> p = gcf().canvas.set_window_title('test of function <sph_harm>')
>>> p = subplot(411);p = title('l=2,m=0');p = imshow(Ylm20.real,cmap=cm.RdBu)
>>> p = subplot(412);p = title('l=2,m=1');p = imshow(Ylm21.real,cmap=cm.RdBu)
>>> p = subplot(413);p = title('l=2,m=2');p = imshow(Ylm22.real,cmap=cm.RdBu)
>>> p = subplot(414);p = title('l=2,m=-2');p = imshow(Ylm2_2.real,cmap=cm.RdBu)
"""
factor = (-1)**m * sqrt( (2*l+1)/(4*pi) * factorial(l-m)/factorial(l+m))
Plm = legendre_(l,m,cos(theta))
return factor*Plm*exp(1j*m*phi)
def sph_harm(theta, phi, l=2, m=1):
"""
Spherical harmonic according to Townsend, 2002.
This function is memoized: once a spherical harmonic is computed, the
result is stored in memory
>>> theta,phi = mgrid[0:pi:20j,0:2*pi:40j]
>>> Ylm20 = sph_harm(theta,phi,2,0)
>>> Ylm21 = sph_harm(theta,phi,2,1)
>>> Ylm22 = sph_harm(theta,phi,2,2)
>>> Ylm2_2 = sph_harm(theta,phi,2,-2)
>>> p = figure()
>>> p = gcf().canvas.set_window_title('test of function <sph_harm>')
>>> p = subplot(411);p = title('l=2,m=0');p = imshow(Ylm20.real,cmap=cm.RdBu)
>>> p = subplot(412);p = title('l=2,m=1');p = imshow(Ylm21.real,cmap=cm.RdBu)
>>> p = subplot(413);p = title('l=2,m=2');p = imshow(Ylm22.real,cmap=cm.RdBu)
>>> p = subplot(414);p = title('l=2,m=-2');p = imshow(Ylm2_2.real,cmap=cm.RdBu)
"""
factor = (-1)**m * sqrt(
(2 * l + 1) / (4 * pi) * factorial(l - m) / factorial(l + m))
Plm = legendre_(l, m, cos(theta))
return factor * Plm * exp(1j * m * phi)
def convex_hull_vol(self, ind, g):
# Compute the convex hull of the region
try:
tri = Delaunay(ind)
except:
# Just triangulate bounding box
mins = ind.min(axis=0)
maxes = ind.max(axis=0)
maxes[maxes==mins] += 1
ind = array(list(itertools.product(*tuple(array([mins,maxes]).T))))
tri = Delaunay(ind)
vol = 0
for simplex in tri.vertices:
pts = tri.points[simplex].T
pts = pts - repeat(pts[:,0][:,newaxis], pts.shape[1],axis=1)
pts = pts[:,1:]
vol += abs(1/float(factorial(pts.shape[0])) * det(pts))
return vol,tri
def convex_hull_vol(self, ind, g):
# Compute the convex hull of the region
try:
tri = Delaunay(ind)
except:
# Just triangulate bounding box
mins = ind.min(axis=0)
maxes = ind.max(axis=0)
maxes[maxes == mins] += 1
ind = np.array(list(it.product(*tuple(np.array([mins, maxes]).T))))
tri = Delaunay(ind)
vol = 0
for simplex in tri.vertices:
pts = tri.points[simplex].T
pts = pts - np.repeat(
pts[:, 0][:, np.newaxis], pts.shape[1], axis=1)
pts = pts[:, 1:]
vol += abs(1 / float(factorial(pts.shape[0])) * det(pts))
return vol, tri
def norm(m, l):
return sqrt((2 * l + 1) / 2. * factorial(l - m) / factorial(l + m))
def simplex(d):
# compute the simplex volume using coordinates
D = np.ones((d.shape[0] + 1, d.shape[1]))
D[1:, :] = d
vol = np.abs(np.linalg.det(D)) / factorial(d.shape[1] - 1)
return vol
else:
hdat[ep][m][n]=atof(item)
ep+=1
file.close() # close the coefficients file
for n in range(1,11): # extrapolate over the next epoch
for m in range(n+1):
gdat[epochs-1][m][n]=gdat[epochs-2][m][n]
+5.*gdat[epochs-1][m][n]
hdat[epochs-1][m][n]=hdat[epochs-2][m][n]
+5.*hdat[epochs-1][m][n]
# ------ declare and initialize fixed parameters for all epochs ---------
a=6371.2 # igrf earth radius
[m,n]=mgrid[0:11,0:11] # set up 11X11 meshgrid
from scipy.misc.common import factorial
"""
build up the "schmidt" coefficients !!! careful with this definition
"""
schmidt=sqrt(2*factorial(n-m)/factorial(n+m))*(-1)**m
schmidt[0,:]=1.
#
#
#fig=figure()
#imshow(schmidt,interpolation='nearest')
#colorbar()
#title("schmidt_mn")
#fig.show()
for item in dat[3:]:
if coeff=="g":
gdat[ep][m][n]=atof(item)
else:
hdat[ep][m][n]=atof(item)
ep+=1
file.close() # close the coefficients file
for n in range(1,11): # extrapolate over the next epoch
for m in range(n+1):
gdat[epochs-1][m][n]=gdat[epochs-2][m][n]
+5.*gdat[epochs-1][m][n]
hdat[epochs-1][m][n]=hdat[epochs-2][m][n]
+5.*hdat[epochs-1][m][n]
# ------ declare and initialize fixed parameters for all epochs ---------
a=6371.2 # igrf earth radius
[m,n]=mgrid[0:11,0:11] # set up 11X11 meshgrid
from scipy.misc.common import factorial
schmidt=sqrt(2*factorial(n-m)/factorial(n+m))*(-1)**m
schmidt[0,:]=1. # build up the "schmidt" coefficients
#!!! careful with this definition
#
#fig=figure()
#imshow(schmidt,interpolation='nearest')
#colorbar()
#title("schmidt_mn")
#fig.show()
def simplex(d): # compute the simplex volume using coordinates D = np.ones((d.shape[0]+1, d.shape[1])) D[1:,:] = d vol = np.abs(np.linalg.det(D)) / factorial(d.shape[1] - 1) return vol
