def Combinations(values, k):
"""This function outputs all the possible combinations of k elements from the vector values"""
if int(k) < 0:
raise ValueError("k must a positive integer")
#Make input vectors column vectors
if values.shape == (1, values.size):
values = sp.atleast2d(values).T.copy()
out = sp.array([]).reshape(0, 1)
n = max(values.shape)
if k == 1:
out = values
else:
#the following loop interates through all the elements of the vector values that have at least k elements after them. For each element it then calls Combinations(values[i+1:], k-1) which returns combinations of size k-1 for the elements succeeding the current element. This is so that we do not get repeats of combinations
#nck = sp.misc.comb(n,k, exact=True)
#out = sp.zeros((nck, k))
for i in range(n - (k - 1)):
#Calculate the number of possible combinations (to allow proper concatenation in the recursive call
nCombs = int(
sp.factorial(n - i) /
(sp.factorial(k - 1) * sp.factorial(n - i - (k - 1))))
#This is the recursive call
print Combinations(values[i + 1:], k - 1).reshape((-1, 1))
out = sp.r_[out, sp.c_[values[i] * sp.ones((nCombs, 1)),
Combinations(values[i + 1:], k -
1).reshape(-1, 1)]]
return out
def _sch_lpmv(n, x):
'''
Outputs array of Schmidt Seminormalized Associated Legendre Functions S_{n}^{m} for m<=n.
Parameters
----------
n : int
Degree of polynomial.
x : float
Point at which to evaluate
Returns
-------
array of values for Legendre functions.
'''
from scipy.special import lpmv
sch = array([1.0])
sch2 = array([(-1.0)**m * sqrt((2.0 * factorial(n - m)) / factorial(n + m))
for m in range(1, n + 1)])
sch = append(sch, sch2)
if isinstance(x, float) or len(x) == 1:
leg = lpmv(arange(0, n + 1), n, x)
return array([sch * leg]).T
else:
for j in range(0, len(x)):
leg = lpmv(range(0, n + 1), n, x[j])
if j == 0:
out = array([sch * leg]).T
else:
out = append(out, array([sch * leg]).T, axis=1)
return out
def Combinations(values, k):
"""This function outputs all possible combinations of k elements from the column vector values"""
n = len(values)
try:
values = sp.row_stack(values)
except:
raise ValueError, "I need a 2d column array"
if k > n:
raise ValueError, "k must be <= %d" % n
elif k <= 0 or k % 1 != 0:
raise ValueError, "k must be > 0"
#out = sp.array([],ndmin=2)
if k == 1:
return values
else:
#This loop iterates through all the elements of the values that have at least
#k elements. For each element it then calls Combinations(values[i+1:], k-1) which
#returns combinations of size k-1 for the elements succeeding the current element
#We do not want to get repeats of combinations
#print "for i in range(%d)" % (n-(k-1))
for i in range(n - (k - 1)):
#Calculate the number of possible combinations (to allow proper concatenation
#in the recursive call
numCombs = sp.factorial(n - i) / (sp.factorial(k - 1) *
sp.factorial(n - i - (k - 1)))
combs = Combinations(values[i:], k - 1)
ones = values[i] * sp.ones((numCombs, 1))
#print "ones: %s \t\t combs: %s" % (str(ones.shape), str(combs.shape))
print combs
def prehamiltonian( genlags, EC, EJ, EL ):
### NB: omits a factor of EJ (on top of the flux dependency and the
### diagonal terms) for use as a derivative later
size = len(genlags)
hbar_w0 = sqrt( 8. * EL * EC )
phi0 = ( 8. * EC / EL ) ** .25
arg = phi0**2/2
genlags = [[ f(arg) for f in row ] for row in genlags]
ret = [ range(size) for i in range(size) ] #values set below
for row in range(size):
for col in range(size):
#the nonzero cosine elements
if (col-row)%2==0:
n = min(row,col)
m = abs(col-row)/2 # because of Hermitianness
ret[row][col] = -(-2)**-m \
* sqrt(factorial(n)/factorial(n+2*m)) \
* phi0**(2*m) * exp(phi0**2/-4) \
* genlags[n][2*m]
#the nonzero sine elements
else:
### IS THIS PART RIGHT?
n = min(row,col)
m = (abs(col-row)-1)/2
ret[row][col] = -(-2)**(-m) * 2**-.5 \
* sqrt(factorial(n)/factorial(n+2*m+1)) \
* phi0**(2*m+1) * exp(phi0**2/-4) \
* genlags[n][2*m+1] ## Check overall signs
return array(ret)
def alpha(k, m, n):
tau = t_intervals[n - 1]
i = np.arange(m + 1)[:, np.newaxis]
the_sum = np.sum((1j*k*omega)**(-m+i-1) * \
tau[:,np.newaxis]**i / factorial(i), axis=-2)
integral = -factorial(m) * np.exp(-1j * k * omega * tau) * the_sum
return integral[:, 1] - integral[:, 0]
def alpha(k,m,n):
tau = t_intervals[n-1]
i = np.arange(m+1)[:,np.newaxis]
the_sum = np.sum((1j*k*omega)**(-m+i-1) * \
tau[:,np.newaxis]**i / factorial(i), axis=-2)
integral = -factorial(m) * np.exp(-1j * k * omega * tau) * the_sum
return integral[:,1] - integral[:,0]
def _sch_lpmv(n,x):
'''
Outputs array of Schmidt Seminormalized Associated Legendre Functions S_{n}^{m} for m<=n.
Parameters
----------
n : int
Degree of polynomial.
x : float
Point at which to evaluate
Returns
-------
array of values for Legendre functions.
'''
from scipy.special import lpmv
sch=array([1.0])
sch2=array([(-1.0)**m*sqrt((2.0*factorial(n-m))/factorial(n+m)) for m in range(1,n+1)])
sch=append(sch,sch2)
if isinstance(x,float) or len(x)==1:
leg=lpmv(arange(0,n+1),n,x)
return array([sch*leg]).T
else:
for j in range(0,len(x)):
leg=lpmv(range(0,n+1),n,x[j])
if j==0:
out=array([sch*leg]).T
else:
out=append(out,array([sch*leg]).T,axis=1)
return out
def Combinations(values, k):
"""This function outputs all possible combinations of k elements from the column vector values"""
n = len(values)
try:
values = sp.row_stack(values)
except:
raise ValueError, "I need a 2d column array"
if k > n:
raise ValueError, "k must be <= %d" % n
elif k<=0 or k%1 != 0:
raise ValueError, "k must be > 0"
#out = sp.array([],ndmin=2)
if k == 1:
return values
else:
#This loop iterates through all the elements of the values that have at least
#k elements. For each element it then calls Combinations(values[i+1:], k-1) which
#returns combinations of size k-1 for the elements succeeding the current element
#We do not want to get repeats of combinations
#print "for i in range(%d)" % (n-(k-1))
for i in range(n-(k-1)):
#Calculate the number of possible combinations (to allow proper concatenation
#in the recursive call
numCombs = sp.factorial(n-i)/(sp.factorial(k-1)*sp.factorial(n-i-(k-1)))
combs = Combinations(values[i:], k-1)
ones = values[i]*sp.ones((numCombs,1))
#print "ones: %s \t\t combs: %s" % (str(ones.shape), str(combs.shape))
print combs
def Combinations(values, k):
"""This function outputs all the possible combinations of k elements from the vector values"""
if int(k) < 0:
raise ValueError("k must a positive integer")
#Make input vectors column vectors
if values.shape == (1,values.size):
values = sp.atleast2d(values).T.copy()
out = sp.array([]).reshape(0,1)
n = max(values.shape)
if k == 1:
out = values
else:
#the following loop interates through all the elements of the vector values that have at least k elements after them. For each element it then calls Combinations(values[i+1:], k-1) which returns combinations of size k-1 for the elements succeeding the current element. This is so that we do not get repeats of combinations
#nck = sp.misc.comb(n,k, exact=True)
#out = sp.zeros((nck, k))
for i in range(n-(k-1)):
#Calculate the number of possible combinations (to allow proper concatenation in the recursive call
nCombs = int(sp.factorial(n-i)/(sp.factorial(k-1)*sp.factorial(n-i-(k-1))))
#This is the recursive call
print Combinations(values[i+1:], k-1).reshape((-1,1))
out = sp.r_[out, sp.c_[values[i]*sp.ones((nCombs,1)), Combinations(values[i+1:], k-1).reshape(-1,1)]]
return out
def delta(a,b,c):
""" Calculate delta """
fac = zeros(4,long)
fac[0] = factorial(a+b-c)
fac[1] = factorial(a-b+c)
fac[2] = factorial(-a+b+c)
fac[3] = factorial(a+b+c+1)
return sqrt(prod(fac[0:3])/fac[3]);
def basis2d(n0,n1,beta=[1.,1.]):
"""2d dimensionless Cartesian basis function"""
b=hermite2d(n0,n1)
b[0]*=((2**n0)*(n.pi**(.5))*scipy.factorial(n0))**(-.5)
exp0=lambda x: beta[0] * b[0](x) * n.exp(-.5*(x**2))
b[1]*=((2**n1)*(n.pi**(.5))*scipy.factorial(n1))**(-.5)
exp1=lambda x: beta[1] * b[1](x) * n.exp(-.5*(x**2))
return [exp0,exp1]
def bernstein(n):
""" Bernstein-Polynome (n!/(i!(n-i)!)* t^i (1-t)^(n-i)"""
t = linspace(0,1,20)
bmat = zeros((20,n+1))
for i in range(0,n+1):
bmat[:,i] = sp.factorial(n)/(sp.factorial(i)*sp.factorial(n-i))*t**i*(1-t)**(n-i)
return bmat
def dimBasis2d(n0,n1,beta=[1.,1.],phs=[1.,1.]):
"""2d dimensional Cartesian basis function of characteristic size beta
phs: additional phase factor, used in the Fourier Transform"""
b=hermite2d(n0,n1)
b[0]*=(beta[0]**(-.5))*(((2**n0)*(n.pi**(.5))*scipy.factorial(n0))**(-.5))
exp0=lambda x: b[0](x/beta[0]) * n.exp(-.5*((x/beta[0])**2)) * phs[0]
b[1]*=(beta[1]**(-.5))*(((2**n1)*(n.pi**(.5))*scipy.factorial(n1))**(-.5))
exp1=lambda x: b[1](x/beta[1]) * n.exp(-.5*((x/beta[1])**2)) * phs[1]
return [exp0,exp1]
def C(n, r):
if n - r > r:
num = np.prod(np.arange(n - r + 1, n+1))
den = factorial(r)
return num / den
else:
num = np.prod(np.arange(r + 1, n+1))
den = factorial(n - r)
return num / den
def plot_messung(n, t, b_len, y_lim, lens, rate):
mu, sigma = np.mean(n), np.std(n)
print "Mittelwert: %.3f" % mu
print "Varianz: %.3f Poisson: %.3f" % (sigma ** 2, mu)
print "Standardabweichung: %.3f Poisson: %.3f" % (sigma, np.sqrt(mu))
print "Standardabweichung des Mittelwerts: %.3f Poisson: %.3f" % (
sigma / np.sqrt(len(n)),
np.sqrt(mu) / np.sqrt(len(n)),
)
print "Gesamtanzahl der Ereignisse:", np.sum(n)
print "mittlere Zaehlrate: %.3f 1/s" % (np.mean(n) / (rate))
print "Standardabweichung der Zaehlrate: %.3f 1/s" % (np.std(n) / rate)
print "Standardabweichung der mittleren Zaehlrate: %.3f 1/s" % (np.std(n) / rate / np.sqrt(len(n)))
print "Schiefe: %.3f Poisson: %.3f" % (np.mean((n - n.mean()) ** 3) / sigma ** 3, 1 / np.sqrt(mu))
print "Kurtosis: %.3f Poisson: %.3f" % (np.mean((n - n.mean()) ** 4) / sigma ** 4 - 3, 1 / (mu))
print ""
fig = plt.figure(figsize=(16, 12))
ax = fig.add_subplot(111)
# the histogram of the data
n, bins, patches = ax.hist(n, lens, normed=1, facecolor="yellow", alpha=0.75)
bincenters = 0.5 * (bins[1:] + bins[:-1])
b = np.linspace(0, b_len, 1000)
b2 = np.arange(0, b_len, 1) + 0.5
# add a 'best fit' line for the normal PDF
# y = mlab.normpdf( b, mu, sigma)
# y2 =mlab.normpdf( b2, mu, sigma)
# l = ax.plot(b, y, 'b--', linewidth=1)
poisson = lambda k: 1.0 / (sc.factorial(k) * np.exp(mu)) * mu ** k
poisson2 = lambda k: 1.0 / (sc.factorial(k) * np.exp(mu + sigma)) * (mu + sigma) ** k
poisson3 = lambda k: 1.0 / (sc.factorial(k) * np.exp(mu - sigma)) * (mu - sigma) ** k
normal_k = lambda k: 1.0 / np.sqrt(2 * np.pi * mu) * np.exp(-(k - mu) ** 2 / (2 * mu))
normal_k2 = (
lambda k: 1.0 / np.sqrt(2 * np.pi * (mu + sigma)) * np.exp(-(k - (mu + sigma)) ** 2 / (2 * (mu + sigma)))
)
normal_k3 = (
lambda k: 1.0 / np.sqrt(2 * np.pi * (mu - sigma)) * np.exp(-(k - (mu - sigma)) ** 2 / (2 * (mu - sigma)))
)
nk = ax.plot(b, normal_k(b), "g--", linewidth=1)
nk = ax.plot(b, normal_k2(b), "g--", linewidth=1)
nk = ax.plot(b, normal_k3(b), "g--", linewidth=1)
p = ax.plot(b, poisson(b), "r--", linewidth=1)
p = ax.plot(b, poisson2(b), "r--", linewidth=1)
p = ax.plot(b, poisson3(b), "r--", linewidth=1)
l = ax.scatter(b2, normal_k(b2), marker="x", c="b")
p = ax.scatter(b2, poisson(b2), marker="x", c="b")
ax.set_xlabel("Anzahl der Impulse")
ax.set_ylabel("Wahrscheinlichkeit")
plt.xlim(0, b_len)
plt.ylim(0, y_lim)
ax.grid(True)
plt.show()
def factorialQuotient(numerator, denominator):
"""
result=numerator!/(denominator!)
"""
diff = numerator - denominator
if diff > 0:
result = factorial(diff)
else:
result = 1.0 / factorial(-diff)
return result
def plot_pdf(order, N, iterations): order_stats = [] for it in range(iterations): numbers = [np.random.uniform(0,1) for i in range(N)] numbers.sort() order_stats.append(numbers[order-1]) plt.figure() n, bins, patches = plt.hist(order_stats, iterations/20, normed=1, facecolor='green') y = lambda x: int(sp.factorial(N))/(int(sp.factorial(N-order))*int(sp.factorial(order-1))) * x**(order-1) * (1-x)**(N-order) plt.plot(bins, y(bins), 'r--', linewidth=3)
def Wigner6j(j1,j2,j3,J1,J2,J3):
#======================================================================
# Calculating the Wigner6j-Symbols using the Racah-Formula
# Author: Ulrich Krohn
# Date: 13th November 2009
#
# Based upon Wigner3j.m from David Terr, Raytheon
# Reference: http://mathworld.wolfram.com/Wigner6j-Symbol.html
#
# Usage:
# from wigner import Wigner6j
# WignerReturn = Wigner6j(j1,j2,j3,J1,J2,J3)
#
# / j1 j2 j3 \
# < >
# \ J1 J2 J3 /
#
#======================================================================
# Check that the js and Js are only integer or half integer
if ( ( 2*j1 != round(2*j1) ) | ( 2*j2 != round(2*j2) ) | ( 2*j2 != round(2*j2) ) | ( 2*J1 != round(2*J1) ) | ( 2*J2 != round(2*J2) ) | ( 2*J3 != round(2*J3) ) ):
print 'All arguments must be integers or half-integers.'
return -1
# Check if the 4 triads ( (j1 j2 j3), (j1 J2 J3), (J1 j2 J3), (J1 J2 j3) ) satisfy the triangular inequalities
if ( ( abs(j1-j2) > j3 ) | ( j1+j2 < j3 ) | ( abs(j1-J2) > J3 ) | ( j1+J2 < J3 ) | ( abs(J1-j2) > J3 ) | ( J1+j2 < J3 ) | ( abs(J1-J2) > j3 ) | ( J1+J2 < j3 ) ):
print '6j-Symbol is not triangular!'
return 0
# Check if the sum of the elements of each traid is an integer
if ( ( 2*(j1+j2+j3) != round(2*(j1+j2+j3)) ) | ( 2*(j1+J2+J3) != round(2*(j1+J2+J3)) ) | ( 2*(J1+j2+J3) != round(2*(J1+j2+J3)) ) | ( 2*(J1+J2+j3) != round(2*(J1+J2+j3)) ) ):
print '6j-Symbol is not triangular!'
return 0
# Arguments for the factorials
t1 = j1+j2+j3
t2 = j1+J2+J3
t3 = J1+j2+J3
t4 = J1+J2+j3
t5 = j1+j2+J1+J2
t6 = j2+j3+J2+J3
t7 = j1+j3+J1+J3
# Finding summation borders
tmin = max(0, max(t1, max(t2, max(t3,t4))))
tmax = min(t5, min(t6,t7))
tvec = arange(tmin,tmax+1,1)
# Calculation the sum part of the 6j-Symbol
WignerReturn = 0
for t in tvec:
WignerReturn += (-1)**t*factorial(t+1)/( factorial(t-t1)*factorial(t-t2)*factorial(t-t3)*factorial(t-t4)*factorial(t5-t)*factorial(t6-t)*factorial(t7-t) )
# Calculation of the 6j-Symbol
return WignerReturn*sqrt( TriaCoeff(j1,j2,j3)*TriaCoeff(j1,J2,J3)*TriaCoeff(J1,j2,J3)*TriaCoeff(J1,J2,j3) )
def multivariate_polya_vectorized(x,alpha):
"""Multivariate Pólya PDF. Vectorized implementation.
"""
x = np.atleast_1d(x)
alpha = np.atleast_1d(alpha)
assert(x.size==alpha.size)
N = x.sum()
A = alpha.sum()
likelihood = factorial(N) / factorial(x).prod() * gamma(A) / gamma(N + A)
likelihood *= (gamma(x + alpha) / gamma(alpha)).prod()
return likelihood
def multivariate_polya(x, alpha):
"""Multivariate Pólya PDF. Basic implementation.
"""
x = np.atleast_1d(x).flatten()
alpha = np.atleast_1d(alpha).flatten()
assert(x.size==alpha.size)
N = x.sum()
A = alpha.sum()
likelihood = factorial(N) * gamma(A) / gamma(N + A)
# likelihood = gamma(A) / gamma(N + A)
for i in range(len(x)):
likelihood /= factorial(x[i])
likelihood *= gamma(x[i] + alpha[i]) / gamma(alpha[i])
return likelihood
def plot_messung3():
H = np.array([6.45 * 10 ** 2, 3.45 * 10 ** 2, 9.5 * 10, 14, 2])
mu = np.sum(H * np.arange(0, 5, 1)) / np.sum(H)
n = np.arange(0, 5, 1)
sigma = np.sqrt((np.sum(H * (np.arange(0, 5, 1) - mu) ** 2) / (np.sum(H) - 1)))
print "Mittelwert: %.3f" % mu
print "Varianz: %.3f Poisson: %.3f" % (sigma ** 2, mu)
print "Standardabweichung: %.3f Poisson: %.3f" % (sigma, np.sqrt(mu))
print "Standardabweichung des Mittelwerts: %.3f Poisson: %.3f" % (
sigma / np.sqrt(np.sum(H)),
np.sqrt(mu) / np.sqrt(np.sum(H)),
)
print "Gesamtanzahl der Ereignisse:", np.sum(H * n)
print "mittlere Zaehlrate: %.3f 1/s" % (mu / (0.5))
print "Standardabweichung der Zaehlrate: %.3f 1/s" % (sigma / 0.5)
print "Standardabweichung der mittleren Zaehlrate: %.3f 1/s" % (sigma / 0.5 / np.sqrt(np.sum(H)))
print "Schiefe: %.3f Poisson: %.3f" % ((np.sum(H * (n - mu) ** 3) / np.sum(H)) / sigma ** 3, 1 / np.sqrt(mu))
print "Kurtosis: %.3f Poisson: %.3f" % ((np.sum(H * (n - mu) ** 4) / np.sum(H)) / sigma ** 4 - 3, 1 / (mu))
print ""
fig = plt.figure(figsize=(16, 12))
plt.scatter(np.arange(0, 4 + 1, 1), H / 1101, marker="^", s=80)
poisson = lambda k: 1.0 / (sc.factorial(k) * np.exp(mu)) * mu ** k
poisson_min = lambda k: 1.0 / (sc.factorial(k) * np.exp(mu - sigma)) * (mu - sigma) ** k
poisson_max = lambda k: 1.0 / (sc.factorial(k) * np.exp(mu + sigma)) * (mu + sigma) ** k
normal_k = lambda k: 1.0 / np.sqrt(2 * np.pi * mu) * np.exp(-(k - mu) ** 2 / (2 * mu))
normal_k_min = (
lambda k: 1.0 / np.sqrt(2 * np.pi * (mu - sigma)) * np.exp(-(k - mu - sigma) ** 2 / (2 * mu - 2 * sigma))
)
normal_k_max = (
lambda k: 1.0 / np.sqrt(2 * np.pi * (mu + sigma)) * np.exp(-(k - mu + sigma) ** 2 / (2 * mu + 2 * sigma))
)
b = np.linspace(0, 5, 100)
b2 = np.arange(0, 5, 1)
# plt.plot(b, normal_k(b), 'g--', linewidth=1)
plt.plot(b, normal_k_min(b), "g--", linewidth=1)
# plt.plot(b, normal_k_max(b), 'g--', linewidth=1)
# plt.plot(b, poisson(b), 'r--', linewidth=1)
plt.plot(b, poisson_min(b), "r--", linewidth=1)
# plt.plot(b, poisson_max(b), 'r--', linewidth=1)
plt.scatter(b2, normal_k(b2), marker="x", c="b")
plt.scatter(b2, poisson(b2), marker="x", c="b")
plt.title("Messung 3")
plt.xlabel("Anzahl der Pulse")
plt.ylabel("Wahrscheinlichkeit")
plt.grid(True)
plt.xlim(0, 5)
plt.ylim(0, 0.7)
plt.show()
def _wigner_laguerre(rho, xvec, yvec, g, parallel):
"""
Using Laguerre polynomials from scipy to evaluate the Wigner function for
the density matrices :math:`|m><n|`, :math:`W_{mn}`. The total Wigner
function is calculated as :math:`W = \sum_{mn} \\rho_{mn} W_{mn}`.
"""
M = prod(rho.shape[0])
X, Y = meshgrid(xvec, yvec)
A = 0.5 * g * (X + 1.0j * Y)
W = zeros(np.shape(A))
# compute wigner functions for density matrices |m><n| and
# weight by all the elements in the density matrix
B = 4 * abs(A) ** 2
if sp.isspmatrix_csr(rho.data):
# for compress sparse row matrices
if parallel:
iterator = (
(m, rho, A, B) for m in range(len(rho.data.indptr) - 1))
W1_out = parfor(_par_wig_eval, iterator)
W += sum(W1_out)
else:
for m in range(len(rho.data.indptr) - 1):
for jj in range(rho.data.indptr[m], rho.data.indptr[m + 1]):
n = rho.data.indices[jj]
if m == n:
W += real(rho[m, m] * (-1) ** m * genlaguerre(m, 0)(B))
elif n > m:
W += 2.0 * real(rho[m, n] * (-1) ** m *
(2 * A) ** (n - m) *
sqrt(factorial(m) / factorial(n)) *
genlaguerre(m, n - m)(B))
else:
# for dense density matrices
B = 4 * abs(A) ** 2
for m in range(M):
if abs(rho[m, m]) > 0.0:
W += real(rho[m, m] * (-1) ** m * genlaguerre(m, 0)(B))
for n in range(m + 1, M):
if abs(rho[m, n]) > 0.0:
W += 2.0 * real(rho[m, n] * (-1) ** m *
(2 * A) ** (n - m) *
sqrt(factorial(m) / factorial(n)) *
genlaguerre(m, n - m)(B))
return 0.5 * W * g ** 2 * np.exp(-B / 2) / pi
def Negbin(x, r, p):
"""
Negative Binomial Log-Likelihood
:param x: data
:param r:
:param p:
>>> Negbin([2,3],6,0.3)
-9.16117424315
"""
x = array(x)
like = sum(r * log(p) + x * log(1 - p) + log(scipy.factorial(x + r - 1)) - log(scipy.factorial(x)) - log(
scipy.factorial(r - 1)))
return like
def Binomial(x, n, p):
"""
Binomial Log-Likelihood
:param x: data
:param n:
:param p:
>>> Binomial([2,3],6,0.3)
-2.81280615454
"""
x = array(x)
like = sum(x * log(p) + (n - x) * log(1. - p) + log(scipy.factorial(n)) - log(scipy.factorial(x)) - log(
scipy.factorial(n - x)))
return like
def polarDimBasis(n0, m0, beta=1.0, phs=1.0):
"""Polar dimensional basis function based on Laguerre polynomials of characteristic size beta
phs: additional phase factor, used in the Fourier Transform"""
b0 = laguerre(n0, m0)
norm = (((-1.0) ** ((n0 - n.abs(m0)) / 2)) / (beta ** (n.abs(m0) + 1))) * (
(float(scipy.factorial(int((n0 - n.abs(m0)) / 2))) / float(scipy.factorial(int((n0 + n.abs(m0)) / 2)))) ** 0.5
)
exp0 = (
lambda r, th: norm
* r ** (n.abs(m0))
* b0((r ** 2.0) / (beta ** 2.0))
* n.exp(-0.5 * (r ** 2.0) / (beta ** 2.0))
* n.exp(-1j * m0 * th)
)
return exp0
def _wigner_laguerre(rho, xvec, yvec, g, parallel):
"""
Using Laguerre polynomials from scipy to evaluate the Wigner function for
the density matrices :math:`|m><n|`, :math:`W_{mn}`. The total Wigner
function is calculated as :math:`W = \sum_{mn} \\rho_{mn} W_{mn}`.
"""
M = prod(rho.shape[0])
X, Y = meshgrid(xvec, yvec)
A = 0.5 * g * (X + 1.0j * Y)
W = zeros(np.shape(A))
# compute wigner functions for density matrices |m><n| and
# weight by all the elements in the density matrix
B = 4 * abs(A)**2
if sp.isspmatrix_csr(rho.data):
# for compress sparse row matrices
if parallel:
iterator = ((m, rho, A, B)
for m in range(len(rho.data.indptr) - 1))
W1_out = parfor(_par_wig_eval, iterator)
W += sum(W1_out)
else:
for m in range(len(rho.data.indptr) - 1):
for jj in range(rho.data.indptr[m], rho.data.indptr[m + 1]):
n = rho.data.indices[jj]
if m == n:
W += real(rho[m, m] * (-1)**m * genlaguerre(m, 0)(B))
elif n > m:
W += 2.0 * real(rho[m, n] * (-1)**m *
(2 * A)**(n - m) *
sqrt(factorial(m) / factorial(n)) *
genlaguerre(m, n - m)(B))
else:
# for dense density matrices
B = 4 * abs(A)**2
for m in range(M):
if abs(rho[m, m]) > 0.0:
W += real(rho[m, m] * (-1)**m * genlaguerre(m, 0)(B))
for n in range(m + 1, M):
if abs(rho[m, n]) > 0.0:
W += 2.0 * real(rho[m, n] * (-1)**m * (2 * A)**(n - m) *
sqrt(factorial(m) / factorial(n)) *
genlaguerre(m, n - m)(B))
return 0.5 * W * g**2 * np.exp(-B / 2) / pi
def zernike2taylor(n, m):
"""
Returns the 2D taylor polynomial, that represents the given zernike
polynomial
**ARGUMENTS**
n,m n and m orders of the Zernike polynomials
**RETURN VALUE**
poly2d instance containing the polynomial
"""
TC = zeros((n + 1, n + 1))
mm = (n - m) / 2
am = abs(m)
if m > 0:
B = mm
if n % 2 == 0:
p = 1
q = (m / 2) - 1
else:
p = 1
q = (m - 1) / 2
else:
B = n - mm
if n % 2 == 0:
p = 0
q = -(m / 2)
else:
p = 0
q = -(m + 1) / 2
for i in range(q + 1):
for j in range(B + 1):
for k in range(B - j + 1):
c = pow(-1, i + j) * binomial(am, 2 * i + p) * binomial(
B - j, k) * int(factorial(n - j)) / (
int(factorial(j)) * int(factorial(mm - j)) *
int(factorial(n - mm - j)))
x_pow = 2 * (i + k) + p
y_pow = n - 2 * (i + j + k) - p
TC[y_pow, x_pow] = TC[y_pow, x_pow] + c
n = TC.shape[0] - 1
cohef = [0.] * ord2i(n)
for i in range(ord2i(n)):
px, py = i2pxpy(i)
cohef[i] = TC[px, py]
return poly2d(cohef)
def PNbarplot(npoints=10000, N=2):
lx = list(np.arange(npoints)*15.0/npoints)
fig = Figure(figsize=(6,6))
# Create a canvas and add the figure to it.
canvas = FigureCanvas(fig)
# Create a subplot.
ax = fig.add_subplot(111)
# Set the title.
ax.set_title('P(N=2, Nbar)',fontsize=14)
# Set the X Axis label.
ax.set_ylabel('P(N=2, Nbar)',fontsize=12)
# Set the Y Axis label.
ax.set_xlabel('Nbar',fontsize=12)
# Display Grid.
ax.grid(True,linestyle='-',color='0.75')
# Generate the Scatter Plot.
ly = [np.exp(-x)*x**(float(N))/sp.factorial(N) for x in lx]
print sum(ly)
ax.scatter(lx,ly,s=.05,color='tomato');
canvas.print_figure('hw4PNbarplot',dpi=500)
def logfact(n):
if memo.has_key(n):
return memo[n]
else:
memo[n] = math.log(scipy.factorial(n))
print "(memoized logfact(%d) = %f)" % (n, memo[n])
return memo[n]
def shapelet_image(basis, beta, centre, hc, nx, ny, size):
""" Takes basis, beta, centre (2-tuple), hc matrix, x, y, size and returns the image of the shapelet of
order nx,ny on an image of size size. Does what getcartim.f does in fBDSM. nx,ny -> 0-nmax
Centre is by Python convention, for retards who count from zero. """
from math import sqrt, pi
try:
from scipy import factorial
except ImportError:
try:
from scipy.misc.common import factorial
except ImportError:
from scipy.misc import factorial
hcx = hc[nx, :]
hcy = hc[ny, :]
ind = N.array([nx, ny])
fact = factorial(ind)
dumr1 = N.sqrt((2.0**(ind)) * sqrt(pi) * fact)
x = (N.arange(size[0], dtype=float) - centre[0]) / beta
y = (N.arange(size[1], dtype=float) - centre[1]) / beta
dumr3 = N.zeros(size[0])
for i in range(size[0]):
for j in range(ind[0] + 1):
dumr3[i] += hcx[j] * (x[i]**j)
B_nx = N.exp(-0.50 * x * x) * dumr3 / dumr1[0] / sqrt(beta)
dumr3 = N.zeros(size[1])
for i in range(size[1]):
for j in range(ind[1] + 1):
dumr3[i] += hcy[j] * (y[i]**j)
B_ny = N.exp(-0.50 * y * y) * dumr3 / dumr1[1] / sqrt(beta)
return N.outer(B_nx, B_ny)
def get_volume(el, nd):
dim = nd.shape[1]
nnd = el.shape[1]
etype = '%d_%d' % (dim, nnd)
if etype == '2_4' or etype == '3_8':
el = elems_q2t(el)
vol = 0.0
bc = nm.zeros((dim, ), dtype=nm.double)
mtx = nm.ones((dim + 1, dim + 1), dtype=nm.double)
mul = 1.0 / sc.factorial(dim)
if dim == 3:
mul *= -1.0
for iel in el:
mtx[:, :-1] = nd[iel, :]
ve = mul * nm.linalg.det(mtx)
vol += ve
bc += ve * mtx.sum(0)[:-1] / nnd
bc /= vol
return vol, bc
def get_volume(el, nd):
dim = nd.shape[1]
nnd = el.shape[1]
etype = '%d_%d' % (dim, nnd)
if etype == '2_4' or etype == '3_8':
el = elems_q2t(el)
vol = 0.0
bc = nm.zeros((dim, ), dtype=nm.double)
mtx = nm.ones((dim + 1, dim + 1), dtype=nm.double)
mul = 1.0 / sc.factorial(dim)
if dim == 3:
mul *= -1.0
for iel in el:
mtx[:,:-1] = nd[iel,:]
ve = mul * nm.linalg.det(mtx)
vol += ve
bc += ve * mtx.sum(0)[:-1] / nnd
bc /= vol
return vol, bc
def shapelet_image(basis, beta, centre, hc, nx, ny, size):
""" Takes basis, beta, centre (2-tuple), hc matrix, x, y, size and returns the image of the shapelet of
order nx,ny on an image of size size. Does what getcartim.f does in fBDSM. nx,ny -> 0-nmax
Centre is by Python convention, for retards who count from zero. """
from math import sqrt, pi
try:
from scipy import factorial
except ImportError:
from scipy.misc.common import factorial
hcx = hc[nx, :]
hcy = hc[ny, :]
ind = N.array([nx, ny])
fact = factorial(ind)
dumr1 = N.sqrt((2.0 ** (ind)) * sqrt(pi) * fact)
x = (N.arange(size[0], dtype=float) - centre[0]) / beta
y = (N.arange(size[1], dtype=float) - centre[1]) / beta
dumr3 = N.zeros(size[0])
for i in range(size[0]):
for j in range(ind[0] + 1):
dumr3[i] += hcx[j] * (x[i] ** j)
B_nx = N.exp(-0.50 * x * x) * dumr3 / dumr1[0] / sqrt(beta)
dumr3 = N.zeros(size[1])
for i in range(size[1]):
for j in range(ind[1] + 1):
dumr3[i] += hcy[j] * (y[i] ** j)
B_ny = N.exp(-0.50 * y * y) * dumr3 / dumr1[1] / sqrt(beta)
return N.outer(B_nx, B_ny)
def _par_wig_eval(args):
"""
Private function for calculating terms of Laguerre Wigner function in parfor.
"""
m,rho,A,B=args
W1 = zeros(shape(A))
for jj in range(rho.data.indptr[m], rho.data.indptr[m+1]):
n = rho.data.indices[jj]
if m == n:
W1 += real(rho[m,m] * (-1)**m * genlaguerre(m,0)(B))
elif n > m:
W1 += 2.0 * real(rho[m,n] * (-1)**m * (2*A)**(n-m) * \
sqrt(factorial(m)/factorial(n)) * genlaguerre(m,n-m)(B))
return W1
def f(t):
args = [t-j1-j2-j3, t-j1-J2-J3, t-J1-j2-J3,
t-J1-J2-j3, j1+j2+J1+J2-t, j2+j3+J2+J3-t,
j3+j1+J3+J1-t]
args = factorial(array(args))
result = args.prod()
return result
def _series(A, Ts, n=10):
"""Matrix power series"""
S = np.eye(A.shape[0]) * Ts
Am = np.matrix(A)
for i in xrange(1, n):
S += (Am ** i) * Ts ** (i + 1) / scipy.factorial(i + 1)
return S
def _series(A, Ts, n=10):
"""Matrix power series"""
S = np.eye(A.shape[0]) * Ts
Am = np.matrix(A)
for i in xrange(1, n):
S += (Am**i) * Ts**(i + 1) / scipy.factorial(i + 1)
return S
def _build_pc_features(self, X):
"""Build polynomial features from X up to self.degree
Assume Xj = [c1, c2, c3, ..., cD] with cI as the Ith feature of the jth
element.
Then build polynomial feature list, e.g. for self.degree=2:
Pj = [1, c1, c2, ..., cD, c1*c1, c1*c2, ..., cD*cD]
Parameters
----------
X : array-like, shape = [n_samples, n_features]
Training vectors, where n_samples is the number of samples
and n_features is the number of features.
Returns
-------
P : ndarray, shape = [n_samples, n_polynomials]
Polynomial features.
n_polynomials = (n_features+self.degree)! /
(n_features! * self.degree!)
"""
N, D = X.shape
# number of polynomials
numP = lambda n, p: (sp.factorial(n + p) /
(sp.factorial(n) * sp.factorial(p)))
# build up index list to combine features, -1 indicates unused feature
I = np.zeros((numP(D, self.degree), self.degree), dtype=int) - 1
for i in range(1, I.shape[0]):
I[i, :] = I[i - 1, :]
for j in range(self.degree):
if I[i - 1, j] + 1 < D:
I[i, j] = I[i - 1, j] + 1
break
j -= 1
while j >= 0:
I[i, j] = I[i, j + 1]
j -= 1
# use index list to build combined polynomial features P
P = np.ones((N, numP(D, self.degree)))
for i in range(I.shape[0]):
for d in range(self.degree):
if I[i, d] > -1:
P[:, i] = P[:, i] * X[:, I[i, d]]
return P
def __init__(self, xi, yi):
"""Construct an interpolator passing through the specified points
The polynomial passes through all the pairs (xi,yi). One may additionally
specify a number of derivatives at each point xi; this is done by
repeating the value xi and specifying the derivatives as successive
yi values.
Parameters
----------
xi : array-like, length N
known x-coordinates
yi : array-like, N by R
known y-coordinates, interpreted as vectors of length R,
or scalars if R=1
Example
-------
To produce a polynomial that is zero at 0 and 1 and has
derivative 2 at 0, call
>>> KroghInterpolator([0,0,1],[0,2,0])
"""
self.xi = np.asarray(xi)
self.yi = np.asarray(yi)
if len(self.yi.shape) == 1:
self.vector_valued = False
self.yi = self.yi[:, np.newaxis]
elif len(self.yi.shape) > 2:
raise ValueError("y coordinates must be either scalars or vectors")
else:
self.vector_valued = True
n = len(xi)
self.n = n
nn, r = self.yi.shape
if nn != n:
raise ValueError(
"%d x values provided and %d y values; must be equal" %
(n, nn))
self.r = r
c = np.zeros((n + 1, r))
c[0] = yi[0]
Vk = np.zeros((n, r))
for k in xrange(1, n):
s = 0
while s <= k and xi[k - s] == xi[k]:
s += 1
s -= 1
Vk[0] = yi[k] / float(factorial(s))
for i in xrange(k - s):
assert xi[i] != xi[k]
if s == 0:
Vk[i + 1] = (c[i] - Vk[i]) / (xi[i] - xi[k])
else:
Vk[i + 1] = (Vk[i + 1] - Vk[i]) / (xi[i] - xi[k])
c[k] = Vk[k - s]
self.c = c
def _par_wig_eval(args):
"""
Private function for calculating terms of Laguerre Wigner function in
parfor.
"""
m, rho, A, B = args
W1 = zeros(np.shape(A))
for jj in range(rho.data.indptr[m], rho.data.indptr[m + 1]):
n = rho.data.indices[jj]
if m == n:
W1 += real(rho[m, m] * (-1)**m * genlaguerre(m, 0)(B))
elif n > m:
W1 += 2.0 * real(rho[m, n] * (-1)**m * (2 * A)**(n - m) * sqrt(
factorial(m) / factorial(n)) * genlaguerre(m, n - m)(B))
return W1
def func_duri(x,q,gamma0,delta,alfa,n=101):
gn=resize(0*x,[n,len(x)])
alfa=abs(cos(alfa))
for k in range(n):
P_k=(exp(-abs(gamma0)*x)*(abs(gamma0)*x)**k)/factorial(k)
gn[k,:]= P_k*exp(-(q*abs(delta)*k**abs(alfa))**2)
g1=sum(gn,axis=0)
return g1
def poiss_prob(x, count):
x = array(x)
P = zeros(x.shape)
i = 0
for xx in x:
P[i] = count**xx * exp(-count) / s.factorial(xx)
i = i + 1
return P
def sixj(j1,j2,j3,l1,l2,l3):
""" Calculate the Wigner six-j symbol of six angular momenta
"""
def bad_values(j1,j2,j3,l1,l2,l3):
""" Check triangular rules for supplied values """
if (j1<(abs(j2-j3)) or j1>(j2+j3)):
return 1
if (j1<(abs(l2-l3)) or j1>(l2+l3)):
return 1
if (l1<(abs(j2-l3)) or l1>(j2+l3)):
return 1
if (l1<(abs(l2-j3)) or l1>(l2+j3)):
return 1
return 0
def delta(a,b,c):
""" Calculate delta """
fac = zeros(4,long)
fac[0] = factorial(a+b-c)
fac[1] = factorial(a-b+c)
fac[2] = factorial(-a+b+c)
fac[3] = factorial(a+b+c+1)
return sqrt(prod(fac[0:3])/fac[3]);
if bad_values(j1,j2,j3,l1,l2,l3):
return 0
jphase=(-1)**(j1+j2+l1+l2);
proddelt=delta(j1,j2,j3)*delta(l1,l2,j3)*delta(l1,j2,l3)*delta(j1,l2,l3);
val = zeros(7,long)
val[0] = j1+j2+l1+l2+1
val[1] = j1+j2-j3
val[2] = l1+l2-j3
val[3] = j1+l2-l3
val[4] = l1+j2-l3
val[5] = -j1-l1+j3+l3
val[6] = -j2-l2+j3+l3
kmax = min(val[0:5])
kmin = max([0, -val[5], -val[6]])
jsum = 0
for k in range(kmin,kmax+1):
jsfac = zeros(8,long)
jsfac[0] = factorial(val[0]-k);
jsfac[1] = factorial(k);
jsfac[2] = factorial(val[1]-k);
jsfac[3] = factorial(val[2]-k);
jsfac[4] = factorial(val[3]-k);
jsfac[5] = factorial(val[4]-k);
jsfac[6] = factorial(val[5]+k);
jsfac[7] = factorial(val[6]+k);
jsum += (-1)**k * jsfac[0] / prod(jsfac[1:])
return jphase*proddelt*jsum
def taylor(p):
r"""
Return the Taylor polynomial of the exponential, up to order p.
"""
from scipy import factorial
coeffs = 1. / factorial(np.arange(p + 1))
return np.poly1d(coeffs[::-1])
def kuiper_FPP(D,N):
"""Compute the false positive probability for the Kuiper statistic
Uses the set of four formulas described in Paltani 2004; they report the resulting function never underestimates the false positive probability but can be a bit high in the N=40..50 range. (They quote a factor 1.5 at the 1e-7 level.
"""
if D<0. or D>2.:
raise ValueError("Must have 0<=D<=2")
if D<2./N:
return 1. - factorial(N)*(D-1./N)**(N-1)
elif D<3./N:
k = -(N*D-1.)/2.
r = sqrt(k**2 - (N*D-2.)/2.)
a, b = -k+r, -k-r
return 1. - factorial(N-1)*(b**(N-1.)*(1.-a)-a**(N-1.)*(1.-b))/float(N)**(N-2)*(b-a)
elif (D>0.5 and N%2==0) or (D>(N-1.)/(2.*N) and N%2==1):
def T(t):
y = D+t/float(N)
return y**(t-3)*(y**3*N-y**2*t*(3.-2./N)/N-t*(t-1)*(t-2)/float(N)**2)
s = 0.
# NOTE: the upper limit of this sum is taken from Stephens 1965
for t in xrange(int(floor(N*(1-D)))+1):
term = T(t)*binomial(N,t)*(1-D-t/float(N))**(N-t-1)
s += term
return s
else:
z = D*sqrt(N)
S1 = 0.
term_eps = 1e-12
abs_eps = 1e-100
for m in itertools.count(1):
T1 = 2.*(4.*m**2*z**2-1.)*exp(-2.*m**2*z**2)
so = S1
S1 += T1
if abs(S1-so)/(abs(S1)+abs(so))<term_eps or abs(S1-so)<abs_eps:
break
S2 = 0.
for m in itertools.count(1):
T2 = m**2*(4.*m**2*z**2-3.)*exp(-2*m**2*z**2)
so = S2
S2 += T2
if abs(S2-so)/(abs(S2)+abs(so))<term_eps or abs(S1-so)<abs_eps:
break
return S1 - 8*D/(3.*sqrt(N))*S2
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 log_factorial(n, N_max=200):
"""
log_factorial(n, N_max = 50)
return Stirling approximation of log of factorial of n if n > N_max
"""
# n = asarray(n)
if n > N_max:
return n * log(n) - n + 0.5 * log(
2 * pi * n) + 1. / 12 / n - 1. / 360 / (n**3)
else:
return log(factorial(n))
def rnm(n, m, rho):
"""
Return an array with the zernike Rnm polynomial calculated at rho points.
**ARGUMENTS:**
=== ==========================================
n n order of the Zernike polynomial
m m order of the Zernike polynomial
rho Matrix containing the radial coordinates.
=== ==========================================
.. note:: For rho>1 the returned value is 0
.. note:: Values for rho<0 are silently returned as rho=0
"""
if (type(n) is not int):
raise Exception("n must be integer")
if (type(m) is not int):
raise Exception("m must be integer")
if (n - m) % 2 != 0:
raise Exception("n-m must be even")
if abs(m) > n:
raise Exception("The following must be true |m|<=n")
mask = where(rho <= 1, False, True)
if (n == 0 and m == 0):
return masked_array(data=ones(rho.shape), mask=mask)
rho = where(rho < 0, 0, rho)
Rnm = zeros(rho.shape)
S = (n - abs(m)) / 2
for s in range(0, S + 1):
CR=pow(-1,s)*factorial(n-s)/ \
(factorial(s)*factorial(-s+(n+abs(m))/2)* \
factorial(-s+(n-abs(m))/2))
p = CR * pow(rho, n - 2 * s)
Rnm = Rnm + p
return masked_array(data=Rnm, mask=mask)
def approximate_taylor_polynomial(f,x,degree,scale,order=None):
"""Estimate the Taylor polynomial of f at x by polynomial fitting
A polynomial
Parameters
----------
f : callable
The function whose Taylor polynomial is sought. Should accept
a vector of x values.
x : scalar
The point at which the polynomial is to be evaluated.
degree : integer
The degree of the Taylor polynomial
scale : scalar
The width of the interval to use to evaluate the Taylor polynomial.
Function values spread over a range this wide are used to fit the
polynomial. Must be chosen carefully.
order : integer or None
The order of the polynomial to be used in the fitting; f will be
evaluated order+1 times. If None, use degree.
Returns
-------
p : poly1d
the Taylor polynomial (translated to the origin, so that
for example p(0)=f(x)).
Notes
-----
The appropriate choice of "scale" is a tradeoff - too large and the
function differs from its Taylor polynomial too much to get a good
answer, too small and roundoff errors overwhelm the higher-order terms.
The algorithm used becomes numerically unstable around order 30 even
under ideal circumstances.
Choosing order somewhat larger than degree may improve the higher-order
terms.
"""
if order is None:
order=degree
n = order+1
# Choose n points that cluster near the endpoints of the interval in
# a way that avoids the Runge phenomenon. Ensure, by including the
# endpoint or not as appropriate, that one point always falls at x
# exactly.
xs = scale*np.cos(np.linspace(0,np.pi,n,endpoint=n%1)) + x
P = KroghInterpolator(xs, f(xs))
d = P.derivatives(x,der=degree+1)
return np.poly1d((d/factorial(np.arange(degree+1)))[::-1])
def EvalWellFunction(self, u, rB):
'''Evaluates the Hantush-Jacob well function.
u dimensionless time
rB dimensionless radius'''
#return zero for large u to avoid numerical issues
if u > 14:
return 0.0
#set number of terms used in summation. as u increases more
#terms are needed
if u <= 2:
endmember = 10
elif u <= 7:
endmember = 20
else:
endmember = 30
#compute temporary variable
r4B = rB**2 / 4.0
#compute summation term
last_term = 0.0
for i in range(1,endmember):
for j in range(1,i+1):
last_term += (-1)**(i+j) * spy.factorial(i - j + 1) / \
(spy.factorial(i+2))**2 * u**(i-j) * r4B**j
#for rB values close to 0 one term is dropped to avoid numerical
#problems
if abs(sci.iv(0.0,rB) - 1.0) < 1e-15:
WellFuncHJ = 2.0 * sci.kv(0.0,rB) \
- sci.iv(0.0,rB) * sci.expn(1,r4B / u) \
+ mth.exp(-1.0 * r4B / u) * (0.5772156649015328606 \
+ mth.log(u) + sci.expn(1,u) - u - u**2 * last_term)
else:
#full expression
WellFuncHJ = 2.0 * sci.kv(0.0,rB) \
- sci.iv(0.0,rB) * sci.expn(1,r4B / u) \
+ mth.exp(-1.0 * r4B / u) * (0.5772156649015328606 \
+ mth.log(u) + sci.expn(1,u) - u + u * (sci.iv(0.0,rB) - 1.0) \
/ r4B - u**2 * last_term)
return WellFuncHJ
def ispline(n,dtk,t):
'''Uniform integrated b-splines
n - Order
dtk - Spacing
t - Time vector'''
x = (t/dtk)+n+1
b = np.zeros(len(x))
for k in range(n+2):
m = x-k-(n+1)/2
up = m**(n+1)
b += ((-1)**k)*nCk(n+1,k)*up*(m>=0)
b = b*dtk/((n+1.0)*factorial(n))
return b
def bspline(n,dtk,t):
'''Uniform b-splines.
n - Order
dtk - Spacing
t - Time vector'''
x = (t/dtk) + n +1
b = np.zeros(len(x))
for k in range(n+2):
m = x-k-(n+1)/2
up = np.power(m,n)
b = b+((-1)**k)*nCk(n+1,k)*up*(m>=0)
b = b/(1.0*factorial(n))
return b
def expSS(x, i=100):
"""Calculate e**x by scaling and squaring"""
from scipy import factorial
results = []
for val in x:
(t, r) = (0, 0)
val = float(val)
while val >= 10:
val = val / 2
t += 2.0
# calculate sum
results.append(sum([val**z / factorial(z) for z in range(i)])**t)
return results
def _qfunc_pure(psi, alpha_mat):
"""
Calculate the Q-function for a pure state.
"""
n = prod(psi.shape)
if isinstance(psi, Qobj):
psi = psi.full().flatten()
else:
psi = psi.T
qmat = abs(
polyval(
fliplr([psi / sqrt(factorial(arange(n)))])[0],
conjugate(alpha_mat)))**2
return real(qmat) * exp(-abs(alpha_mat)**2) / pi
def qfunc1(psi, alpha_mat):
"""
private function used by qfunc
"""
n = prod(psi.shape)
if isinstance(psi, Qobj):
psi = array(psi.trans().full())[0, :]
else:
psi = psi.T
qmat1 = abs(
polyval(
fliplr([psi / sqrt(factorial(arange(0, n)))])[0],
conjugate(alpha_mat)))**2
qmat1 = real(qmat1) * exp(-abs(alpha_mat)**2) / pi
return qmat1
def _qfunc_pure(psi, alpha_mat):
"""
Calculate the Q-function for a pure state.
"""
n = prod(psi.shape)
if isinstance(psi, Qobj):
psi = array(psi.trans().full())[0, :]
else:
psi = psi.T
qmat1 = abs(
polyval(
fliplr([psi / sqrt(factorial(arange(0, n)))])[0],
conjugate(alpha_mat)))**2
qmat1 = real(qmat1) * exp(-abs(alpha_mat)**2) / pi
return qmat1
def Stehfest(self, t):
"""
Stehfest method works for smooth function
f(t) = ln(2)/t \sum_{i=1}^{N}V_i*F(u),
where u = (i*ln(2))/t is the Laplace variable,
and
V_i = (-1)^(N/2 + i)*sum_int((i+1)/2)^min(i, N/2) K^(N/2 +1)*(2K)! /(N/2-K)!*K!*(K-1)!*(i-K)!*(2K-i)!,
in which N is even number, 8, 10, or 12, (or 16).
@param t is time
@param F is the laplace image function
"""
F = self.F
N = 10
N2 = N / 2
ln2 = np.log(2)
G = sci.factorial(np.array(range(0, N + 1, 1))) # factorial 0!...N!
V = np.zeros_like(range(0, N + 1, 1))
sign = 1
if N2 % 2:
sign = -1
for i in range(1, N + 1, 1):
kmin = (i + 1) / 2
kmax = np.minimum(i, N2)
sign = -sign
for K in range(kmin, kmax + 1, 1):
V[i] += np.power(K,N2)*G[2*K]/\
(G[N/2-K]*G[K]*G[K-1]*G[i-K]*G[2*K-i])
V[i] = sign * V[i]
#print "V[1..N] = ", V #debug
f = np.zeros_like(t)
for j in range(0, len(t), 1):
ln2t = ln2 / t[j]
u = 0
for i in range(1, N + 1, 1):
u += ln2t # u = i*ln2/t
f[j] += V[i] * F(u)
f[j] = ln2t * f[j]
return f
def perm_unique_trans(n, startk=0, verbose=False):
"""
Generate unique permutations of n numbers, with the constraint
that none of the transitions can be the same. Continues to generate
sequences until all the possible permutations have been tested.
This is not a terribly fast algorithm.
"""
from scipy import factorial
out = []
seqpairs = []
maxk = factorial(n, exact=True)
k = startk
while k < maxk:
perm = perm_dfr(perm_rr(n, k))
k += 1
permpairs = set("%d%d" % (perm[i], perm[i + 1]) for i in range(len(perm) - 1))
pairmatches = [len(x.intersection(permpairs)) for x in seqpairs]
if len(pairmatches) == 0 or max(pairmatches) == 0:
out.append(perm)
seqpairs.append(permpairs)
yield k, perm
elif verbose and k % 10000 == 0:
print("Checked to permutation %d" % k)
