def BK(xip, nr):
r"""Return BK_nr."""
if np.isrealobj(xip):
# To keep it real in Laplace-domain [exp(-1j*0) = 1-0j].
return special.kve(nr, xip)
else:
return np.exp(-1j * np.imag(xip)) * special.kve(nr, xip)
def Rlam(x, lam=None):
v1 = kve(lam + 1, x) * np.exp(x)
v0 = kve(lam, x) * np.exp(x)
val = v1 / v0
if np.isinf(v1) or np.isinf(v0) or v0 == 0 or v1 == 0:
lv1 = np.log(sympy.Float(mpmath.besselk(np.abs(lam + 1), x)))
lv0 = np.log(sympy.Float(mpmath.besselk(np.abs(lam), x)))
val = np.exp(lv1 - lv0)
return val
def gigGH(x=None, par=None, invS=None):
"""
Calculate expectation of W, 1/W and log(W) (E-Step)
W is assumed to follow Generalized Inverse Gaussian W~GIG(psi, chi, lambda)
Refer to p9 A Mixture of Generalized Hyperbolic Distributions
Params:
a, v: numerical values
b : numpy 2d array
# Calculate a, b, c - expectations of w, 1/w, log(w) given x
# w : a = a_ig = E[W_ig | x_i, Z_ig = 1]
# 1/w : b = b_ig = E[1 / W_ig | x_i, Z_ig = 1]
# log(w) : c = c_ig = E[log W_ig | x_i, Z_ig = 1]
# abc is a MATRIX
# returns a matrix with dim length(a) x 3 stima
# a = omega_g + (beta_g)' * sigma_inverse * beta_g
# b = omega + delta(x_i, mu_g | sigma_inverse)
# kv1 is the denominator BesselK
# kv is the numerator BesselK
Input
x: 2d array
data
par: ClusterParam object
Return:
tuple contains 3 values: E[W], E[1/W] and E[log(W)]
"""
if invS is None:
invS = np.linalg.inv(par.sigma)
omega = par.omega
a = omega + np.matmul(np.matmul(par.alpha, invS), par.alpha)
delta = np.array(
[mahalanobis(x[i, :], par.mu, invS)**2 for i in range(x.shape[0])])
b = omega + delta
v = par.lamda - par.mu.shape[0] / 2
sab = np.sqrt(a * b)
kvv = kve(v + 1, sab) / kve(v, sab)
sb_a = np.sqrt(b / a)
# Expected value of w and 1/w
w = kvv * sb_a
invw = kvv / sb_a - 2 * v / b
logw = np.log(sb_a) + richardson_gradient(log_besselKv,
x=v * np.ones(sab.shape),
y=sab,
eps=1e-8,
d=0.0001,
r=6,
v=2)
return (w, invw, logw)
def Fm(x, m, r1, r2, semi=True):
ra = r1
rb = r2
if (r2 < r1):
rb = r1
ra = r2
if m > 20 and semi:
za = x * ra
zb = x * rb
if (za < 1e-5) and (zb < 1e-5):
e = -1 + m * math.log(ra / rb * (m + 1.0) / m) - 0.5 * math.log(
(m + 1.0) * m) - math.log(rb / 2 / (m + 1.0))
return -math.exp(2 * e) / 4.0
#return 0.0
sqa1 = math.sqrt(m**2 + za**2)
sqa2 = math.sqrt((m + 1)**2 + za**2)
sqb1 = math.sqrt(m**2 + zb**2)
sqb2 = math.sqrt((m + 1)**2 + zb**2)
li1 = sqa1 + m * math.log(za / (m + sqa1)) - 0.5 * math.log(sqa1)
li2 = sqa2 + (m + 1) * math.log(za /
(m + 1 + sqa2)) - 0.5 * math.log(sqa2)
lk1 = -sqb1 - m * math.log(zb / (m + sqb1)) - 0.5 * math.log(sqb1)
lk2 = -sqb2 - (m + 1) * math.log(zb /
(m + 1 + sqb2)) - 0.5 * math.log(sqb2)
L = 2.0 * (li1 + lk1)
Li = 2.0 * (li2 - li1)
Lk = 2.0 * (lk2 - lk1)
return math.exp(L) * x * x * (math.exp(Li) - 1) * (math.exp(Lk) -
1) / 4
#return math.exp(L) * (math.exp(Li) - 1) * (math.exp(Lk) - 1)/4
im1 = special.ive(m, x * ra)
km1 = special.kve(m, x * rb)
im2 = special.ive(m + 1, x * ra)
km2 = special.kve(m + 1, x * rb)
e = math.exp(x * (ra - rb))
q1 = im1 * km1 * im1 * km1
q2 = im2 * km2 * im2 * km2
q3 = im1 * km2 * im1 * km2
#q4 = q3
q4 = im2 * km1 * im2 * km1
#q = x * (im1 * km1 - im2 * km2) * e
#print r1, r2, m, e, q
return x * x * (im1 * im1 - im2 * im2) * (km2 * km2 - km1 * km1) * e * e
def dKo(self,n,z):
"""odd second-kind radial Mathieu function derivative (DKo)
(analogous to K Bessel functions, called Gek in McLachlan p248),
for scalar or vector orders or argument
n: order (scalar or vector)
z: radial argument (scalar or vector)"""
from scipy.special import kve,ive
n,z,sqrtq,enz,epz,v1,v2,M,EV,OD = self._RadDerivFuncSetup(n,z)
dy = np.empty((n.shape[0],z.shape[0]),dtype=complex)
ord = self.ord[0:M+2]
I = ive(ord[0:M+2,None],v1[None,:])[:,None,:]
K = kve(ord[0:M+2,None],v2[None,:])[:,None,:]
dI = self._deriv(v1,I[0:M+2,0,:],0)[:,None,:]
dK = self._deriv(v2,K[0:M+2,0,:],1)[:,None,:]
j = (n[:]-1)//2
dy[EV,:] = (sqrtq/self.B[0,j[EV],0,:]*np.sum(self.B[0:M,j[EV],0,:] *
(epz*I[0:M,:,:]*dK[2:M+2,:,:] - enz*dI[0:M,:,:]*K[2:M+2,:,:] -
(epz*I[2:M+2,:,:]*dK[0:M,:,:] - enz*dI[2:M+2,:,:]*K[0:M,:,:])),axis=0))
dy[OD,:] = (sqrtq/self.A[0,j[OD],1,:]*np.sum(self.A[0:M,j[OD],1,:] *
(epz*I[0:M,:,:]*dK[1:M+1,:,:] - enz*dI[0:M,:,:]*K[1:M+1,:,:] +
epz*I[1:M+1,:,:]*dK[0:M,:,:] - enz*dI[1:M+1,:,:]*K[0:M,:,:]),axis=0))
dy *= np.exp(np.abs(v1.real) - v2)[None,:]
dy[n==0,:] = np.NaN # dKo_0() invalid
return np.squeeze(dy)
def Ko(self,n,z):
"""odd second-kind radial Mathieu function (Ko)
(analogous to K Bessel functions, called Gek in McLachlan p248),
for scalar or vector orders or argument
n: order (scalar or vector)
z: radial argument (scalar or vector)"""
from scipy.special import kve,ive
n,z,sqrtq,v1,v2,M,EV,OD = self._RadFuncSetup(n,z)
y = np.empty((n.shape[0],z.shape[0]),dtype=complex)
ord = self.ord[0:M+2]
I = ive(ord[0:M+2,None],v1[None,:])[:,None,:]
K = kve(ord[0:M+2,None],v2[None,:])[:,None,:]
j = (n[:]-1)//2
y[EV,:] = (np.sum(self.B[0:M,j[EV],0,:] *
(I[0:M,:,:]*K[2:M+2,:,:] - I[2:M+2,:,:]*K[0:M,:,:]),axis=0)/
self.B[0,j[EV],0,:])
y[OD,:] = (np.sum(self.A[0:M,j[OD],1,:] *
(I[0:M,:,:]*K[1:M+1,:,:] + I[1:M+1,:,:]*K[0:M,:,:]),axis=0)/
self.A[0,j[OD],1,:])
y *= np.exp(np.abs(v1.real) - v2)[None,:]
y[n==0,:] = np.NaN # Ko_0() invalid
return np.squeeze(y)
def Ko(self, n, z):
"""odd second-kind radial Mathieu function (Ko)
(analogous to K Bessel functions, called Gek in McLachlan p248),
for scalar or vector orders or argument
n: order (scalar or vector)
z: radial argument (scalar or vector)"""
from scipy.special import kve, ive
n, z, sqrtq, v1, v2, M, EV, OD = self._RadFuncSetup(n, z)
y = np.empty((n.shape[0], z.shape[0]), dtype=complex)
ord = self.ord[0:M + 2]
I = ive(ord[0:M + 2, None], v1[None, :])[:, None, :]
K = kve(ord[0:M + 2, None], v2[None, :])[:, None, :]
j = (n[:] - 1) // 2
y[EV, :] = (np.sum(self.B[0:M, j[EV], 0, :] *
(I[0:M, :, :] * K[2:M + 2, :, :] -
I[2:M + 2, :, :] * K[0:M, :, :]),
axis=0) / self.B[0, j[EV], 0, :])
y[OD, :] = (np.sum(self.A[0:M, j[OD], 1, :] *
(I[0:M, :, :] * K[1:M + 1, :, :] +
I[1:M + 1, :, :] * K[0:M, :, :]),
axis=0) / self.A[0, j[OD], 1, :])
y *= np.exp(np.abs(v1.real) - v2)[None, :]
y[n == 0, :] = np.NaN # Ko_0() invalid
return np.squeeze(y)
def test_besselk10e(self):
xs = [0.1, 1.0, 10.0]
for x in xs:
sp = kve(10, x)
hal = besselkne(10, x)
relerr = abs((sp - hal) / sp)
self.assertLessEqual(relerr, 0.05)
def testBesselKveZNegativeVLarge(self, dtype, rtol):
seed_stream = test_util.test_seed_stream()
v = np.linspace(100., 200., num=10, dtype=dtype)
z = tf.random.uniform([10], -10., -1., seed=seed_stream(), dtype=dtype)
z, bessel_kve = self.evaluate([z, tfp.math.bessel_kve(v, z)])
scipy_kve = scipy_special.kve(v, z)
self.assertAllClose(bessel_kve, scipy_kve, rtol=rtol)
def get_prior(E_x, E_invx, k, lamb, rho, tau):
#The underlying probability distribution
#P(x)=Gamma(x | k, lambda)
#Q(x)=GIG(x | k, rho, tau)
#The log(x) term is canceled out between logP(x) and logQ(x)
#Initialization
prior = 0
#Avoid "invalid multiplication" error caused by small tau
tau[tau < 1e-100] = 1e-100
#The logP(x) term other than the canceled log(x)
prior = prior + (k * np.log(lamb) - sc.gammaln(k)) * E_x.size
prior = prior - np.sum(lamb * E_x)
#The -logQ(x) term
#log2
prior = prior + np.log(2) * E_x.size
#-k/2*(log(rho)-log(tau))
prior = prior - (k/2) * np.sum(np.log(rho) - np.log(tau))
#rho*x+tau/x
prior = prior + np.sum(rho * E_x + tau * E_invx)
#log(bessel(2*sqrt(rho*tau)))
prior = prior + np.sum(np.log(sc.kve(k, 2*np.sqrt(rho*tau))))
return prior
def ddghypGH(x, par, log=False, invS=None):
mu = par.mu
sigma = par.sigma
alpha = par.alpha
if (len(mu) != len(alpha)):
print('mu and alpha do not have the same length.')
return
d = len(mu)
omega = np.exp(np.log(par.omega))
lamda = par.lamda
if not invS:
invS = np.linalg.pinv(sigma)
pa = omega + np.matmul(np.matmul(alpha, invS), alpha)
delta = np.array(
[mahalanobis(x[i, :], mu, invS)**2 for i in range(x.shape[0])])
mx = omega + delta
kx = np.sqrt(mx * pa)
xmu = x - mu
lvx = np.zeros((x.shape[0], 4))
lvx[:, 0] = (lamda - d / 2) * np.log(kx)
lvx[:, 1] = np.log(kve(lamda - d / 2, kx)) - kx
lvx[:, 2] = np.matmul(np.matmul(xmu, invS), alpha)
lv = np.zeros(6)
if np.log(np.linalg.det(sigma)) is np.inf: # need checking
sigma = np.eye(len(mu))
lv[0] = -1 / 2 * np.log(np.linalg.det(sigma))
lv[0] -= d / 2 * (np.log(2) + np.log(np.pi))
lv[1] = omega - np.log(kve(lamda, omega))
lv[2] = -lamda / 2 * np.log(1)
lv[3] = lamda * np.log(omega) * 0
lv[4] = (d / 2 - lamda) * np.log(pa)
val = np.sum(lvx.transpose(), axis=0) + sum(lv)
if not log:
val = np.exp(val)
return val
def Fm(x, m, r1, r2, semi = True):
ra = r1
rb = r2
if (r2 < r1):
rb = r1
ra = r2
if m > 20 and semi:
za = x * ra
zb = x * rb
if (za < 1e-5) and (zb < 1e-5):
e = -1 + m * math.log(ra/rb * (m + 1.0)/m) -0.5 *math.log((m+1.0)*m) - math.log(rb/2/(m + 1.0))
return -math.exp(2*e)/4.0
#return 0.0
sqa1 = math.sqrt(m**2 + za**2)
sqa2 = math.sqrt((m + 1)**2 + za**2)
sqb1 = math.sqrt(m**2 + zb**2)
sqb2 = math.sqrt((m + 1)**2 + zb**2)
li1 = sqa1 + m * math.log(za/(m + sqa1)) - 0.5 * math.log(sqa1)
li2 = sqa2 + (m + 1) * math.log(za/(m + 1 + sqa2)) - 0.5 * math.log(sqa2)
lk1 = -sqb1 - m * math.log(zb/(m + sqb1)) - 0.5 * math.log(sqb1)
lk2 = -sqb2 - (m + 1)* math.log(zb/(m + 1 + sqb2)) - 0.5 * math.log(sqb2)
L = 2.0*(li1 + lk1)
Li = 2.0*(li2 - li1)
Lk = 2.0*(lk2 - lk1)
return math.exp(L) * x * x * (math.exp(Li) - 1) * (math.exp(Lk) - 1)/4
#return math.exp(L) * (math.exp(Li) - 1) * (math.exp(Lk) - 1)/4
im1 = special.ive(m, x * ra)
km1 = special.kve(m, x * rb)
im2 = special.ive(m + 1, x * ra)
km2 = special.kve(m + 1, x * rb)
e = math.exp(x * (ra - rb))
q1 = im1 * km1 * im1 * km1
q2 = im2 * km2 * im2 * km2
q3 = im1 * km2 * im1 * km2
#q4 = q3
q4 = im2 * km1 * im2 * km1
#q = x * (im1 * km1 - im2 * km2) * e
#print r1, r2, m, e, q
return x * x * (im1 * im1 - im2 * im2) * (km2 * km2 - km1 * km1) * e * e
def estimate(self, context, data):
pdf = ScaleMixture()
alpha = context.prior.alpha
beta = context.prior.beta
d = context._d
if len(data.shape) == 1:
data = data[:, numpy.newaxis]
p = -alpha + 0.5 * d
b = 2 * beta
a = 1e-5 + data.sum(-1) ** 2
s = numpy.clip((numpy.sqrt(b) * kve(p + 1, numpy.sqrt(a * b)))
/ (numpy.sqrt(a) * kve(p, numpy.sqrt(a * b))),
1e-10, 1e10)
pdf.scales = s
context.prior.estimate(s)
pdf.prior = context.prior
return pdf
def get_GIG_expectation(gamma, rho, tau):
#Initialize variables
if np.isscalar(gamma) == True:
gamma = gamma * np.ones_like(rho) #Extend into the same size
E_x = np.zeros_like(rho)
E_invx = np.zeros_like(rho)
#Avoid "invalid multiplication" error caused by small tau
tau[tau < 1e-100] = 1e-100
#Compute the scaled bessel function "kve" using the scipy.special
rt_rho = np.sqrt(rho)
rt_tau = np.sqrt(tau)
bessel_gamma = sc.kve(gamma, 2*rt_rho*rt_tau)
bessel_gamma_1minus = sc.kve(gamma-1, 2*rt_rho*rt_tau)
bessel_gamma_1plus = sc.kve(gamma+1, 2*rt_rho*rt_tau)
#Compute expectations of x and inverse x under the GIG
E_x = (bessel_gamma_1plus * rt_tau) / (rt_rho * bessel_gamma)
E_invx = (bessel_gamma_1minus * rt_rho) / (rt_tau * bessel_gamma)
return E_x, E_invx
def compute_gig_expectations(alpha, beta, gamma):
if np.asarray(alpha).size == 1:
alpha = alpha * np.ones_like(beta)
Ex, Exinv = np.zeros_like(beta), np.zeros_like(beta)
# For very small values of gamma and positive values of alpha, the GIG
# distribution becomes a gamma distribution, and its expectations are both
# cheaper and more stable to compute that way.
gig_inds = (gamma > 1e-200)
gam_inds = (gamma <= 1e-200)
if np.any(alpha[gam_inds] < 0):
raise ValueError("problem with arguments.")
# Compute expectations for GIG distribution.
sqrt_beta = np.sqrt(beta[gig_inds])
sqrt_gamma = np.sqrt(gamma[gig_inds])
# Note that we're using the *scaled* version here, since we're just
# computing ratios and it's more stable.
bessel_alpha_minus = special.kve(alpha[gig_inds] - 1,
2 * sqrt_beta * sqrt_gamma)
bessel_alpha = special.kve(alpha[gig_inds], 2 * sqrt_beta * sqrt_gamma)
bessel_alpha_plus = special.kve(alpha[gig_inds] + 1,
2 * sqrt_beta * sqrt_gamma)
sqrt_ratio = sqrt_gamma / sqrt_beta
Ex[gig_inds] = bessel_alpha_plus * sqrt_ratio / bessel_alpha
Exinv[gig_inds] = bessel_alpha_minus / (sqrt_ratio * bessel_alpha)
# Compute expectations for gamma distribution where we can get away with
# it.
Ex[gam_inds] = alpha[gam_inds] / beta[gam_inds]
Exinv[gam_inds] = beta[gam_inds] / (alpha[gam_inds] - 1)
Exinv[Exinv < 0] = np.inf
return (Ex, Exinv)
def compute_gig_expectations(alpha, beta, gamma):
if np.asarray(alpha).size == 1:
alpha = alpha * np.ones_like(beta)
Ex, Exinv = np.zeros_like(beta), np.zeros_like(beta)
# For very small values of gamma and positive values of alpha, the GIG
# distribution becomes a gamma distribution, and its expectations are both
# cheaper and more stable to compute that way.
gig_inds = (gamma > 1e-200)
gam_inds = (gamma <= 1e-200)
if np.any(alpha[gam_inds] < 0):
raise ValueError("problem with arguments.")
# Compute expectations for GIG distribution.
sqrt_beta = np.sqrt(beta[gig_inds])
sqrt_gamma = np.sqrt(gamma[gig_inds])
# Note that we're using the *scaled* version here, since we're just
# computing ratios and it's more stable.
bessel_alpha_minus = special.kve(alpha[gig_inds] - 1, 2 * sqrt_beta *
sqrt_gamma)
bessel_alpha = special.kve(alpha[gig_inds], 2 * sqrt_beta * sqrt_gamma)
bessel_alpha_plus = special.kve(alpha[gig_inds] + 1, 2 * sqrt_beta *
sqrt_gamma)
sqrt_ratio = sqrt_gamma / sqrt_beta
Ex[gig_inds] = bessel_alpha_plus * sqrt_ratio / bessel_alpha
Exinv[gig_inds] = bessel_alpha_minus / (sqrt_ratio * bessel_alpha)
# Compute expectations for gamma distribution where we can get away with
# it.
Ex[gam_inds] = alpha[gam_inds] / beta[gam_inds]
Exinv[gam_inds] = beta[gam_inds] / (alpha[gam_inds] - 1)
Exinv[Exinv < 0] = np.inf
return (Ex, Exinv)
def k012e(x):
"""Returns k0e(x), k1e(x) and k2e(x) for real or complex x.
For real x, the fast exponentially scaled K_n Bessel functions, k0e end k1e
are defined in scipy.special, but not ke2, which is computed here from the
other two using the recurion:
K_n(x) = K_{n-2}(x) + 2(n-1)/z * K_{n-1}(x)
For complex x, kve(0,x) and kve(1,x) and the recursion are used.
Returns the outputs all three functions, evaluated at x.
"""
if np.iscomplexobj(x):
k0 = special.kve(0,x)
k1 = special.kve(1,x)
k2 = k0 + 2.0*k1/x
else:
k0 = special.k0e(x)
k1 = special.k1e(x)
k2 = k0 + 2.0*k1/x
return k0, k1, k2
#def ddxlogK1e(z):
"""
def logpyt(self, theta, t):
alpha = np.exp(theta['log_alpha'])
lam = np.exp(theta['log_lambda'])
abs_y = np.maximum(np.abs(self.data[t]), tol)
# Numerical issues when computing the modified Bessel function
#print("Value of (y, alpha, lambda) = ({}, {}, {})".format(abs_y, alpha, lam))
#print("Bessel function = {}".format(kv(lam - 0.5, alpha * abs_y)))
if kv(lam - 0.5, alpha * abs_y) > 0:
return 2 * lam * np.log(alpha) + (lam - 0.5) * np.log(abs_y) + \
np.log(kv(lam - 0.5, alpha*abs_y)) - 0.5 * np.log(np.pi) - \
gammaln(lam) - (lam - 0.5)*np.log(2*alpha)
return 2 * lam * np.log(alpha) + (lam - 0.5) * np.log(abs_y) + \
np.log(kve(lam - 0.5, alpha*abs_y)) - abs_y - 0.5 * np.log(np.pi) - \
gammaln(lam) - (lam - 0.5) * np.log(2 * alpha)
def logpyt(self, theta, t):
eta = np.exp(theta['log_eta'])
c = np.exp(theta['log_c'])
alpha = np.sqrt(2 * c)
lam = eta
abs_y = np.maximum(np.abs(self.data[t]), tol)
# Numerical issues when computing the modified Bessel function
if kv(lam - 0.5, alpha * abs_y) > 0:
return 2 * lam * np.log(alpha) + (lam - 0.5) * np.log(abs_y) + \
np.log(kv(lam - 0.5, alpha*abs_y)) - 0.5 * np.log(np.pi) - \
gammaln(lam) - (lam - 0.5)*np.log(2*alpha)
return 2 * lam * np.log(alpha) + (lam - 0.5) * np.log(abs_y) + \
np.log(kve(lam - 0.5, alpha*abs_y)) - abs_y - 0.5 * np.log(np.pi) - \
gammaln(lam) - (lam - 0.5) * np.log(2 * alpha)
def gig_gamma_term(Ex, Exinv, rho, tau, a, b):
score = 0
cut_off = 1e-200
zero_tau = (tau <= cut_off)
non_zero_tau = (tau > cut_off)
score += Ex.size * (a * log(b) - special.gammaln(a))
score -= np.sum((b - rho) * Ex)
score -= np.sum(non_zero_tau) * log(0.5)
score += np.sum(tau[non_zero_tau] * Exinv[non_zero_tau])
score -= 0.5 * a * np.sum(np.log(rho[non_zero_tau]) -
np.log(tau[non_zero_tau]))
# It's numerically safer to use scaled version of besselk
score += np.sum(np.log(special.kve(a, 2 * np.sqrt(rho[non_zero_tau] *
tau[non_zero_tau]))) -
2 * np.sqrt(rho[non_zero_tau] * tau[non_zero_tau]))
score += np.sum(-a * np.log(rho[zero_tau]) + special.gammaln(a))
return score
def gig_gamma_term(Ex, Exinv, rho, tau, a, b):
score = 0
cut_off = 1e-200
zero_tau = (tau <= cut_off)
non_zero_tau = (tau > cut_off)
score += Ex.size * (a * log(b) - special.gammaln(a))
score -= np.sum((b - rho) * Ex)
score -= np.sum(non_zero_tau) * log(0.5)
score += np.sum(tau[non_zero_tau] * Exinv[non_zero_tau])
score -= 0.5 * a * np.sum(
np.log(rho[non_zero_tau]) - np.log(tau[non_zero_tau]))
# It's numerically safer to use scaled version of besselk
score += np.sum(
np.log(
special.kve(a, 2 *
np.sqrt(rho[non_zero_tau] * tau[non_zero_tau]))) -
2 * np.sqrt(rho[non_zero_tau] * tau[non_zero_tau]))
score += np.sum(-a * np.log(rho[zero_tau]) + special.gammaln(a))
return score
def dKo(self, n, z):
"""odd second-kind radial Mathieu function derivative (DKo)
(analogous to K Bessel functions, called Gek in McLachlan p248),
for scalar or vector orders or argument
n: order (scalar or vector)
z: radial argument (scalar or vector)"""
from scipy.special import kve, ive
n, z, sqrtq, enz, epz, v1, v2, M, EV, OD = self._RadDerivFuncSetup(
n, z)
dy = np.empty((n.shape[0], z.shape[0]), dtype=complex)
ord = self.ord[0:M + 2]
I = ive(ord[0:M + 2, None], v1[None, :])[:, None, :]
K = kve(ord[0:M + 2, None], v2[None, :])[:, None, :]
dI = self._deriv(v1, I[0:M + 2, 0, :], 0)[:, None, :]
dK = self._deriv(v2, K[0:M + 2, 0, :], 1)[:, None, :]
j = (n[:] - 1) // 2
dy[EV, :] = (sqrtq / self.B[0, j[EV], 0, :] *
np.sum(self.B[0:M, j[EV], 0, :] *
(epz * I[0:M, :, :] * dK[2:M + 2, :, :] -
enz * dI[0:M, :, :] * K[2:M + 2, :, :] -
(epz * I[2:M + 2, :, :] * dK[0:M, :, :] -
enz * dI[2:M + 2, :, :] * K[0:M, :, :])),
axis=0))
dy[OD, :] = (sqrtq / self.A[0, j[OD], 1, :] * np.sum(
self.A[0:M, j[OD], 1, :] *
(epz * I[0:M, :, :] * dK[1:M + 1, :, :] - enz * dI[0:M, :, :] *
K[1:M + 1, :, :] + epz * I[1:M + 1, :, :] * dK[0:M, :, :] -
enz * dI[1:M + 1, :, :] * K[0:M, :, :]),
axis=0))
dy *= np.exp(np.abs(v1.real) - v2)[None, :]
dy[n == 0, :] = np.NaN # dKo_0() invalid
return np.squeeze(dy)
def Mtil(m, epr, mur, bet, nu, b):
def produto(A,B):
C = _np.zeros((A.shape[0],B.shape[1],A.shape[2]),dtype=complex)
for i in range(A.shape[0]):
for j in range(B.shape[1]):
for k in range(A.shape[1]):
C[i,j,:] = C[i,j,:] + A[i,k,:]*B[k,j,:]
return C
for i in range(len(b)): # lembrando que length(b) = número de camadas - 1
x = nu[i+1,:] * b[i]
y = nu[i ,:] * b[i]
Mt = _np.zeros((4,4,w.shape[0]),dtype=complex)
if i<len(b)-1:
D = _np.zeros((4,4,nu.shape[1]),dtype=complex)
z = nu[i+1,:]*b[i+1]
if not any(z.real<0):
ind = (z.real<60)
A = _scyspe.iv(m,z[ind])
B = _scyspe.kv(m,z[ind])
C = _scyspe.iv(m,x[ind])
E = _scyspe.kv(m,x[ind])
D[0,0,:] = 1
D[1,1,ind] = - B*C/(A*E)
D[1,1,~ind] = - _np.exp(-2*(z[~ind]-x[~ind]))
D[2,2,:] = 1
D[3,3,ind] = - B*C/(A*E)
D[3,3,~ind] = - _np.exp(-2*(z[~ind]-x[~ind]))
Mt[0,0,:] = -nu[i+1,:]**2*b[i]/epr[i+1,:]*(
epr[i+1,:]/nu[i+1,:]*(-_scyspe.kve(m-1,x)/_scyspe.kve(m,x) - m/x)
- epr[i,:]/nu[i,:] *( _scyspe.ive(m-1,y)/_scyspe.ive(m,y) - m/y))
Mt[0,1,:] = -nu[i+1,:]**2*b[i]/epr[i+1,:]*(
epr[i+1,:]/nu[i+1,:]*(-_scyspe.kve(m-1,x)/_scyspe.kve(m,x) - m/x)
- epr[i,:]/nu[i,:] *(-_scyspe.kve(m-1,y)/_scyspe.kve(m,y) - m/y))
Mt[1,0,:] = -nu[i+1,:]**2*b[i]/epr[i+1,:]*(
epr[i+1,:]/nu[i+1,:]*( _scyspe.ive(m-1,x)/_scyspe.ive(m,x) - m/x)
- epr[i,:]/nu[i,:] *( _scyspe.ive(m-1,y)/_scyspe.ive(m,y) - m/y))
Mt[1,1,:] = -nu[i+1,:]**2*b[i]/epr[i+1,:]*(
epr[i+1,:]/nu[i+1,:]*( _scyspe.ive(m-1,x)/_scyspe.ive(m,x) - m/x)
- epr[i,:]/nu[i,:] *(-_scyspe.kve(m-1,y)/_scyspe.kve(m,y) - m/y))
Mt[0,2,:] = (nu[i+1,:]**2/nu[i,:]**2 - 1)*m/(bet*epr[i+1,:])
Mt[0,3,:] = Mt[0,2,:]
Mt[1,2,:] = Mt[0,2,:]
Mt[1,3,:] = Mt[0,2,:]
Mt[2,2,:] = -nu[i+1,:]**2*b[i]/mur[i+1,:]*(
mur[i+1,:]/nu[i+1,:]*(-_scyspe.kve(m-1,x)/_scyspe.kve(m,x) - m/x)
- mur[i,:]/nu[i,:] *( _scyspe.ive(m-1,y)/_scyspe.ive(m,y) - m/y))
Mt[2,3,:] = -nu[i+1,:]**2*b[i]/mur[i+1,:]*(
mur[i+1,:]/nu[i+1,:]*(-_scyspe.kve(m-1,x)/_scyspe.kve(m,x) - m/x)
- mur[i,:]/nu[i,:] *(-_scyspe.kve(m-1,y)/_scyspe.kve(m,y) - m/y))
Mt[3,2,:] = -nu[i+1,:]**2*b[i]/mur[i+1,:]*(
mur[i+1,:]/nu[i+1,:]*( _scyspe.ive(m-1,x)/_scyspe.ive(m,x) - m/x)
- mur[i,:]/nu[i,:] *( _scyspe.ive(m-1,y)/_scyspe.ive(m,y) - m/y))
Mt[3,3,:] = -nu[i+1,:]**2*b[i]/mur[i+1,:]*(
mur[i+1,:]/nu[i+1,:]*( _scyspe.ive(m-1,x)/_scyspe.ive(m,x) - m/x)
- mur[i,:]/nu[i,:] *(-_scyspe.kve(m-1,y)/_scyspe.kve(m,y) - m/y))
Mt[2,0,:] = (nu[i+1,:]**2/nu[i,:]**2 - 1)*m/(bet*mur[i+1,:])
Mt[2,1,:] = Mt[2,0,:]
Mt[3,0,:] = Mt[2,0,:]
Mt[3,1,:] = Mt[2,0,:]
if len(b) == 1:
M = Mt
else:
if (i ==0):
M = produto(D,Mt)
elif i < len(b)-1:
M = produto(D,produto(Mt,M))
else:
M = produto(Mt,M)
return M
def rel_thermal_beta(mu):
# computes thermal velocity/c for mu
return np.sqrt(1.e0 - (kve(1, mu) / kve(2, mu) + 3.e0 / mu)**(-2))
def BK(xip, nr):
"""Return BK_nr."""
return np.exp(-1j * np.imag(xip)) * kve(nr, xip)
def BK(xip, nr):
r"""Return BK_nr."""
return np.exp(-1j * np.imag(xip)) * special.kve(nr, xip)
def besselkve(self, v, x):
if v == 1:
return special.k1e(x)
else:
return special.kve(v, x)
def VerifyBesselKve(self, v, z, rtol):
bessel_kve, v, z = self.evaluate([tfp.math.bessel_kve(v, z), v, z])
scipy_kve = scipy_special.kve(v, z)
self.assertAllClose(bessel_kve, scipy_kve, rtol=rtol)
def log_besselKv(nu, y):
return np.log(kve(nu, y)) - np.log(y)
def Mtil(m, epr, mur, bet, nu, b):
def produto(A, B):
C = _np.zeros((A.shape[0], B.shape[1], A.shape[2]), dtype=complex)
for i in range(A.shape[0]):
for j in range(B.shape[1]):
for k in range(A.shape[1]):
C[i, j, :] = C[i, j, :] + A[i, k, :] * B[k, j, :]
return C
for i in range(
len(b)): # lembrando que length(b) = número de camadas - 1
x = nu[i + 1, :] * b[i]
y = nu[i, :] * b[i]
Mt = _np.zeros((4, 4, w.shape[0]), dtype=complex)
if i < len(b) - 1:
D = _np.zeros((4, 4, nu.shape[1]), dtype=complex)
z = nu[i + 1, :] * b[i + 1]
if not any(z.real < 0):
ind = (z.real < 60)
A = _scyspe.iv(m, z[ind])
B = _scyspe.kv(m, z[ind])
C = _scyspe.iv(m, x[ind])
E = _scyspe.kv(m, x[ind])
D[0, 0, :] = 1
D[1, 1, ind] = -B * C / (A * E)
D[1, 1, ~ind] = -_np.exp(-2 * (z[~ind] - x[~ind]))
D[2, 2, :] = 1
D[3, 3, ind] = -B * C / (A * E)
D[3, 3, ~ind] = -_np.exp(-2 * (z[~ind] - x[~ind]))
Mt[0, 0, :] = -nu[i + 1, :]**2 * b[i] / epr[i + 1, :] * (
epr[i + 1, :] / nu[i + 1, :] *
(-_scyspe.kve(m - 1, x) / _scyspe.kve(m, x) - m / x) -
epr[i, :] / nu[i, :] *
(_scyspe.ive(m - 1, y) / _scyspe.ive(m, y) - m / y))
Mt[0, 1, :] = -nu[i + 1, :]**2 * b[i] / epr[i + 1, :] * (
epr[i + 1, :] / nu[i + 1, :] *
(-_scyspe.kve(m - 1, x) / _scyspe.kve(m, x) - m / x) -
epr[i, :] / nu[i, :] *
(-_scyspe.kve(m - 1, y) / _scyspe.kve(m, y) - m / y))
Mt[1, 0, :] = -nu[i + 1, :]**2 * b[i] / epr[i + 1, :] * (
epr[i + 1, :] / nu[i + 1, :] *
(_scyspe.ive(m - 1, x) / _scyspe.ive(m, x) - m / x) -
epr[i, :] / nu[i, :] *
(_scyspe.ive(m - 1, y) / _scyspe.ive(m, y) - m / y))
Mt[1, 1, :] = -nu[i + 1, :]**2 * b[i] / epr[i + 1, :] * (
epr[i + 1, :] / nu[i + 1, :] *
(_scyspe.ive(m - 1, x) / _scyspe.ive(m, x) - m / x) -
epr[i, :] / nu[i, :] *
(-_scyspe.kve(m - 1, y) / _scyspe.kve(m, y) - m / y))
Mt[0, 2, :] = (nu[i + 1, :]**2 / nu[i, :]**2 -
1) * m / (bet * epr[i + 1, :])
Mt[0, 3, :] = Mt[0, 2, :]
Mt[1, 2, :] = Mt[0, 2, :]
Mt[1, 3, :] = Mt[0, 2, :]
Mt[2, 2, :] = -nu[i + 1, :]**2 * b[i] / mur[i + 1, :] * (
mur[i + 1, :] / nu[i + 1, :] *
(-_scyspe.kve(m - 1, x) / _scyspe.kve(m, x) - m / x) -
mur[i, :] / nu[i, :] *
(_scyspe.ive(m - 1, y) / _scyspe.ive(m, y) - m / y))
Mt[2, 3, :] = -nu[i + 1, :]**2 * b[i] / mur[i + 1, :] * (
mur[i + 1, :] / nu[i + 1, :] *
(-_scyspe.kve(m - 1, x) / _scyspe.kve(m, x) - m / x) -
mur[i, :] / nu[i, :] *
(-_scyspe.kve(m - 1, y) / _scyspe.kve(m, y) - m / y))
Mt[3, 2, :] = -nu[i + 1, :]**2 * b[i] / mur[i + 1, :] * (
mur[i + 1, :] / nu[i + 1, :] *
(_scyspe.ive(m - 1, x) / _scyspe.ive(m, x) - m / x) -
mur[i, :] / nu[i, :] *
(_scyspe.ive(m - 1, y) / _scyspe.ive(m, y) - m / y))
Mt[3, 3, :] = -nu[i + 1, :]**2 * b[i] / mur[i + 1, :] * (
mur[i + 1, :] / nu[i + 1, :] *
(_scyspe.ive(m - 1, x) / _scyspe.ive(m, x) - m / x) -
mur[i, :] / nu[i, :] *
(-_scyspe.kve(m - 1, y) / _scyspe.kve(m, y) - m / y))
Mt[2, 0, :] = (nu[i + 1, :]**2 / nu[i, :]**2 -
1) * m / (bet * mur[i + 1, :])
Mt[2, 1, :] = Mt[2, 0, :]
Mt[3, 0, :] = Mt[2, 0, :]
Mt[3, 1, :] = Mt[2, 0, :]
if len(b) == 1:
M = Mt
else:
if (i == 0):
M = produto(D, Mt)
elif i < len(b) - 1:
M = produto(D, produto(Mt, M))
else:
M = produto(Mt, M)
return M
def k1e(x):
if np.iscomplexobj(x):
return special.kve(1,x)
else:
return special.k1e(x)
def k0e(x):
if np.iscomplexobj(x):
return special.kve(0,x)
else:
return special.k0e(x)