def part_of_sum(k_on, k_off, mu, n, r, t):
part_b1 = math.comb(n, r)
logging.debug(part_b1)
part_b2 = (-1) ** r * sc.poch(-k_off, r) * sc.poch(1 - k_off, n - r) * np.exp(-r * t)
logging.debug(part_b2)
part_b3 = 1 / sc.poch(1 + k_on + k_off, r)
logging.debug(part_b3)
part_b4 = 1 / sc.poch(2 - k_on - k_off, n - r)
logging.debug(part_b4)
part_b5 = sc.hyp1f1(k_off + r,
1 + k_on + k_off + r,
mu * np.exp(-t))
logging.debug(part_b5)
# NB: some terms in the Pochhammer's reappear in 1F1
# and may cancel each other and this may avoid dividing by zero?
part_b6 = sc.hyp1f1(1 - k_off + n - r,
2 - k_on - k_off + n - r,
-mu)
logging.debug(part_b6)
part_b = part_b1 * part_b2 * part_b3 * part_b4 * part_b5 * part_b6
logging.debug(part_b)
return part_b
def hansen_coefficient(s, r, x):
a = poch(3 / 2, r / 2 - 1) * poch(3 / 2 + s, r / 2 - 1) / poch(
s + 1, r / 2 - 1)
b = np.zeros_like(x)
for j in xrange(int(r / 2)):
b = b + (-1)**j * poch(1 / 2 + s + r / 2, j) * x**(
2 * j) / factorial(j) / factorial(r / 2 - 1 - j) / poch(3 / 2, j)
return a * b
def generalized_assoc_laguerre(x, n, k):
sum = np.linspace(0 + 0j, 0 + 0j, len(x))
for i in range(n + 1):
numerator = special.poch(-n, i) * special.poch(i + k + 1, n - i)
denumerator = float(special.factorial(i))
sum += numerator * (x**i) / denumerator
return sum / special.factorial(n)
def keval(self, x, k, order):
return (
(-1) ** (order - k) *
special.poch(-order, k) *
special.poch(-x, k) *
self.alpha ** (order - k) /
math.factorial(k)
)
def _hyp0f2(b1, b2, z, eps=1e-6, nmax=10):
sum = 0
#accumulate the sum from scratch, no convenient identities, but 5 terms seems good enough, use 10 to be safe
#mpmath does this but don't want to introduce dependency just for one function
for k in range(nmax):
sum += 1 / (poch(b1, k) *
poch(b2, k)) * z**k / np.math.factorial(k)
return sum
def p_stationary(n, k_on, k_off, k_syn, k_d):
k_on = k_on / k_d
k_off = k_off / k_d
k_syn = k_syn / k_d
part1 = k_syn**n/fac(n)
part2 = sc.poch(k_on, n)/sc.poch(k_on + k_off, n)
part3 = sc.hyp1f1(k_on + n, k_on + k_off + n, - k_syn)
ret_val = part1 * part2 * part3
return ret_val
def eval(self, x, order):
mp.dps = 25
mp.pretty = True
return (
special.poch(1 - self.N, order) *
hyp3f2(-order, -x, 1 + order, 1, 1 - self.N, 1)
)
def test_fht_exact(n):
rng = np.random.RandomState(3491349965)
# for a(r) a power law r^\gamma, the fast Hankel transform produces the
# exact continuous Hankel transform if biased with q = \gamma
mu = rng.uniform(0, 3)
# convergence of HT: -1-mu < gamma < 1/2
gamma = rng.uniform(-1 - mu, 1 / 2)
r = np.logspace(-2, 2, n)
a = r**gamma
dln = np.log(r[1] / r[0])
offset = fhtoffset(dln, mu, initial=0.0, bias=gamma)
A = fht(a, dln, mu, offset=offset, bias=gamma)
k = np.exp(offset) / r[::-1]
# analytical result
At = (2 / k)**gamma * poch((mu + 1 - gamma) / 2, gamma)
assert_allclose(A, At)
def laplace_b(s, j, n, alpha):
"""
Calculates nth derivative with respect to a (alpha) of Laplace coefficient b_s^j(a).
Uses recursion and scipy special functions.
Arguments
---------
s : float
half-integer parameter of Laplace coefficient.
j : int
integer parameter of Laplace coefficient.
n : int
return nth derivative with respect to a of b_s^j(a)
a : float
semimajor axis ratio a1/a2 (alpha)
"""
assert alpha >= 0 and alpha < 1, "alpha not in range [0,1): alpha={}".format(
alpha)
if j < 0:
return laplace_b(s, -j, n, alpha)
if n >= 2:
return s * (laplace_b(s + 1, j - 1, n - 1, alpha) -
2 * alpha * laplace_b(s + 1, j, n - 1, alpha) +
laplace_b(s + 1, j + 1, n - 1, alpha) - 2 *
(n - 1) * laplace_b(s + 1, j, n - 2, alpha))
if n == 1:
return s * (laplace_b(s + 1, j - 1, 0, alpha) -
2 * alpha * laplace_b(s + 1, j, 0, alpha) +
laplace_b(s + 1, j + 1, 0, alpha))
return 2 * poch(s, j) * alpha**j * hyp2f1(s, s + j, j + 1, alpha**
2) / factorial(j)
def torontonian_analytical(l, nbar):
r"""Return the value of the Torontonian of the O matrices generated by gen_omats
Args:
l (int): number of modes
nbar (float): mean photon number of the first mode (the only one not prepared in vacuum)
Returns:
float: Value of the torontonian of gen_omats(l,nbar)
"""
if np.allclose(l, nbar, atol=1e-14, rtol=0.0):
return 1.0
beta = -(nbar / (l * (1 + nbar)))
pref = factorial(l) / beta
p1 = pref * l / poch(1 / beta, l + 2)
p2 = pref * beta / poch(2 + 1 / beta, l)
return (p1 + p2) * (-1)**l
def check_sph_harm(l, m, theta, phi):
print(sp.sph_harm(m, l, phi, theta))
Clm = np.sqrt(
(2 * l + 1.0) / 4 / np.pi * sp.poch(l + abs(m) + 1, -2 * abs(m)))
[Pmlv, Pmlvdiff] = sp.lpmn(abs(m), l, np.cos(theta))
ans = Clm * Pmlv[abs(m), l] * np.exp(1j * m * phi)
if (m < 0):
ans = (-1)**m * np.conjugate(ans)
print(ans)
def matel(x, i, j):
n = min(i, j)
m = max(i, j)
factor = (-0.5)**((m-n)/2.) \
*1./np.sqrt(special.poch(n+1,m-n)) \
*x**(m-n) \
*np.exp(-0.25*x**2) \
*special.eval_genlaguerre(n, m-n, 0.5*x**2)
return factor
def spherical_harmonic_normalization(n, m, norm='full'):
r"""The normalization factor for real valued spherical harmonics.
.. math::
N_n^m = \sqrt{\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}
Parameters
----------
n : int
The spherical harmonic order.
m : int
The spherical harmonic degree.
norm : 'full', 'semi', optional
Normalization to use. Can be either fully normalzied on the sphere or
semi-normalized.
Returns
-------
norm : double
The normalization factor.
"""
if np.abs(m) > n:
factor = 0.0
else:
if norm == 'full':
z = n + m + 1
factor = _spspecial.poch(z, -2 * m)
factor *= (2 * n + 1) / (4 * np.pi)
if int(m) != 0:
factor *= 2
factor = np.sqrt(factor)
elif norm == 'semi':
z = n + m + 1
factor = _spspecial.poch(z, -2 * m)
if int(m) != 0:
factor *= 2
factor = np.sqrt(factor)
else:
raise ValueError("Unknown normalization.")
return factor
def paris_exponential_series(a, b, z, i, maxiters):
"""The exponentially small addition to the asymptotic expansion on the
negative real axis.
The argument `i` is the truncation index for the original series.
This function does not reproduce all of the significant figures of Table 2
in Paris (2013), suggesting a small implementation bug.
Possibly fewer than all 5 terms should be summed for optimal peformance.
(There are of course infinitely many terms, but the first 5 are the only
ones for which the polynomial coefficients were included in the paper.)
"""
M = 5
theta = a - b
x = -z
if i < maxiters:
# The optimal truncation term has been determined.
v = a + i + theta
else:
# We don't know how many terms are optimal exactly (it's more than the
# number of terms summed), so we'll use an approximation.
v = x
A = np.arange(M)
A = poch(1 - a, A)*poch(b - a, A)/gamma(A + 1)
first_sum = np.sum((-1)**np.arange(M) * A * x**(-np.arange(M)))
second_sum = 0
for idx in xrange(M):
B = sum((-2)**k*poch(0.5, k)*A[idx-k]*np.polyval(PARIS_G[k,:],v - x - (idx-k))*6**(-2*k)
for k in xrange(idx + 1))
second_sum += (-1)**idx * B * x**(-idx)
if np.real(a) and np.real(b) and (b < 0 or a < 0):
c = np.exp(gammaln(b + 0j) - gammaln(a + 0j) - x + theta*np.log(x))
c = np.real(c)
else:
c = np.exp(gammaln(b) - gammaln(a) - x + theta*np.log(x))
return c*(np.cos(np.pi*theta)*first_sum
- 2*np.sin(np.pi*theta)/np.sqrt(2*np.pi*x)*second_sum)
def test_gegenbauer(self):
a = 5 * np.random.random() - 0.5
if np.any(a == 0):
a = -0.2
Ca0 = orth.gegenbauer(0, a)
Ca1 = orth.gegenbauer(1, a)
Ca2 = orth.gegenbauer(2, a)
Ca3 = orth.gegenbauer(3, a)
Ca4 = orth.gegenbauer(4, a)
Ca5 = orth.gegenbauer(5, a)
assert_array_almost_equal(Ca0.c, array([1]), 13)
assert_array_almost_equal(Ca1.c, array([2 * a, 0]), 13)
assert_array_almost_equal(Ca2.c, array([2 * a * (a + 1), 0, -a]), 13)
assert_array_almost_equal(
Ca3.c,
array([4 * sc.poch(a, 3), 0, -6 * a * (a + 1), 0]) / 3.0, 11)
assert_array_almost_equal(
Ca4.c,
array([
4 * sc.poch(a, 4), 0, -12 * sc.poch(a, 3), 0, 3 * a * (a + 1)
]) / 6.0, 11)
assert_array_almost_equal(
Ca5.c,
array([
4 * sc.poch(a, 5), 0, -20 * sc.poch(a, 4), 0,
15 * sc.poch(a, 3), 0
]) / 15.0, 11)
def solution_me(k: int, alpha: float, m: int = 1) -> float:
"""
Functions that return value of probability that was calculated analytical using Master equation and stand for
if alpha!=0
P(k) = bunch of gamma functions...
note that this equation will broke at alpha -> 0, thus if alpha == 0 :
m^(k-m) / (m+1)^(k-m+1)
:param k: vertex degree
:param alpha: alpha value
:param m: at this moment always equal 1, originally this is yet another generalization of BA graph
:return: probability of given vertex degree
"""
if alpha > 0.001:
m = 1
ret = 2 / (2 * m + 2 - alpha * m)
ret *= poch(2 * m / alpha - m, 1 + 2 / alpha)
ret /= poch(2 * m / alpha - 2 * m + k, 1 + 2 / alpha)
else:
m = 1
ret = m**(k - m) / (m + 1)**(k - m + 1)
return ret
def threeFtwo(a, b):
"""
Hypergerometric 3_F_2([a1,a2,a3],[b1,b2],1)
Used in calcluations of KaulaF function
Arguments
---------
a : list of ints
b : list of ints
Returns
-------
float
"""
a1, a2, a3 = a
b1, b2 = b
kmax = min(1 - a1, 1 - a2, 1 - a3)
tot = 0
for k in range(0, kmax):
tot += poch(a1, k) * poch(a2, k) * poch(a3, k) / poch(b1, k) / poch(
b2, k) / factorial(k)
return tot
def get_vecsph(x, y, z, l, m):
m_abs = abs(
m
) #calculate everything with |m|, then at the end conjugate and multiply (-1)^m if needed
A1lm = np.zeros(3, dtype=complex)
A2lm = np.zeros(3, dtype=complex)
A3lm = np.zeros(3, dtype=complex)
if m_abs > l:
print("m>l encountered in get_vecsph")
return A1lm, A2lm, A3lm
r, theta, phi, rhat, thehat, phihat = Cart_to_sphere(x, y, z)
costheta = np.cos(theta)
[Pmlv, Pmlvdiff] = sp.lpmn(
m_abs, l, costheta
) #assembling spherical harmonics by hand since we also need their derivatives
Clm = np.sqrt(
(2 * l + 1.0) / 4 / np.pi *
sp.poch(l + m_abs + 1, -2 * m_abs)) #prefactor for spherical harmonics
phiphase = np.exp(1j * m_abs * phi)
#A3lm = rhat*sp.sph_harm(m,l,phi,theta) #scipy harmonics have azimuthal angle as first angle argument
A3lm = rhat * Clm * Pmlv[m_abs, l] * phiphase
if theta == 0.0 or costheta == -1.0: #special case, avoid 1/sin(theta) nan, see Kristensson appendix D
if m_abs == 1:
prefact = np.sqrt((2 * l + 1.0) / 16 / np.pi)
A1lm = -prefact * np.array([1j, -1.0, 0.0])
A2lm = -prefact * np.array([1.0, 1j, 0.0])
if costheta == -1.0:
A1lm = (-1)**l * A1lm
A2lm = (-1)**l * A2lm
#otherwise nothing is done to A1lm, A2lm, and we get back 0s
elif l > 0: #A100 A200 are all 0
pdvYphi_over_sine = 1j * m_abs * Clm * Pmlv[
m_abs, l] * phiphase / np.sin(theta)
pdvYthe = -np.sin(theta) * Clm * phiphase
pdvYthe *= Pmlvdiff[m_abs, l]
A1lm = (thehat * pdvYphi_over_sine - phihat * pdvYthe) / np.sqrt(
l * (l + 1.0))
A2lm = (thehat * pdvYthe + phihat * pdvYphi_over_sine) / np.sqrt(
l * (l + 1.0))
if m < 0:
A1lm = (-1)**m * np.conjugate(A1lm)
A2lm = (-1)**m * np.conjugate(A2lm)
A3lm = (-1)**m * np.conjugate(A3lm)
return A1lm, A2lm, A3lm
def hansen_coefficient(s, r, x):
a = poch(3/2, r/2-1)*poch(3/2+s, r/2-1)/poch(s+1, r/2-1)
b = np.zeros_like(x)
for j in xrange(int(r/2)):
b = b + (-1)**j*poch(1/2+s+r/2, j)*x**(2*j)/factorial(j)/factorial(r/2-1-j)/poch(3/2, j)
return a*b
def poch_(z, m):
return 1.0 / poch(z, m)
def poch_minus(z, m):
return 1.0 / poch(z, -m)
def poch_(z, m):
return 1.0 / poch(z, m)
def poch_minus(z, m):
return 1.0 / poch(z, -m)
def c_k_partial( k, immi_rate, birth_rate, death_rate, N, K, comp_overlap ):
value = 1.0
if k == 0.0:
return value
else:
""" previous calc.
for i in np.arange(1,k+1):
value = value*( ( immi_rate + birth_rate*(i-1))/( i*(death_rate + (birth_rate-death_rate)*(i*(1-comp_overlap)+comp_overlap*N)/K) ) )
"""
c = ( death_rate*K/((birth_rate-death_rate)) + comp_overlap*N ) / ( 1-comp_overlap )
value = ( birth_rate*K / ( (birth_rate - death_rate)*(1-comp_overlap) ) ) ** k * poch( immi_rate/birth_rate, k ) / ( factorial(k)*poch( c + 1, k ) )
return value
def R(i, immi_rate, birth_rate, death_rate, N, K, comp_overlap):
if i == 1:
return 1.0
else:
return ( birth_rate*K )**(i-1) * poch( immi_rate/birth_rate + 1 , i - 1 ) / ( fact(i-1)*poch(death_rate*K + comp_overlap*N + 1, i-1) )
def negative_binom(minus_q, l):
# scipy.special.binom returns a NaN when called at a
# negative integer so I use this alternate formulation
# when the argument is potenially a negative integer
return (-1)**l * poch(-1 * minus_q, l) / factorial(l)
def falling_factorial(n, p):
return 1.0 / spspec.poch(n + 1, -p)
def arcsin_taylor_factor(p):
"""Returns the (1+2p)th Taylor expansion factor of arcsin(x) around x=0, which is a_p = (1/2)_p / (1+2p)*p!"""
return poch(1 / 2, p) / ((1 + 2 * p) * math.factorial(p))
def pdf_smalln_better_unnormalized(x, stoch, cap, delta=1.):#this creates problems as it divides ~0 by ~0 or something like that
P1 = 1.;
return P1*(1-stoch)/(1-2*stoch)/x*poch(1+cap*(1+delta)/(1-2*stoch),x-1)/poch(1+(1-stoch+cap*delta/2.)/(1-stoch),x-1)*((1-2*stoch)/(1-stoch))**x
def pdf(a, b, n, k):
return sp.poch(n, k) * sp.poch(a, n) * sp.poch(b, k) / (
math.factorial(k) * sp.poch(a + b, n) * sp.poch(n + a + b, k))
def kurtosis(m, omega):
return (m * (4 * m + 1) - 2 * (2 * m + 1) * sp.poch(m, 1 / 2)**2) / (
m - sp.poch(m, 1 / 2)**2)**2 - 3
def skewness(m, omega):
return (sp.poch(m, 1 / 2) * (1 / 2 - 2 *
(m - sp.poch(m, 1 / 2)**2))) / math.pow(
m - sp.poch(m, 1 / 2)**2, 3 / 2)
def poch_sum_ln_15_4_1(a,b,c,z):
m = -a
n = np.arange(m+1)
return np.sum(poch(-m,n)*z**n*np.exp(pochln(b,n)-gammaln(n+1))/poch(c,n))
def poch_sum_15_4_1(a,b,c,z):
m = -a
n = np.arange(m+1)
return np.sum(poch(-m,n)*poch(b,n)*z**n/(poch(c,n)*gamma(n+1)))