def speed_test(self):
'''Over all speed test'''
N = int(2e3)
xmax = 5
x = xmax * (np.random.rand(N) - 0.5)
T = {}
tic = time.perf_counter()
erfcx(x)
T['Benchmark (erfcx)'] = time.perf_counter() - tic
tic = time.perf_counter()
self.int_brute_force(x)
T['Brute force integration'] = time.perf_counter() - tic
tic = time.perf_counter()
self.int_exact(x)
T['Simplified integration'] = time.perf_counter() - tic
tic = time.perf_counter()
self.int_fast(x)
T['Fast approximation'] = time.perf_counter() - tic
rep = [
'Speed Test Result', 'Number of samples: {}'.format(N),
'Sample Range: [-{},{}]'.format(xmax, xmax)
]
rep += ['Time Elapsed | Relative to benchmark']
for k in T:
rep.append('{}: {:.1e} | {:.1e}'.format(
k, T[k], T[k] / T['Benchmark (erfcx)']))
print('\n'.join(rep))
def _bemg(x_in, la, lb, u, s, A):
a = math.exp(la) #la,lb default 2
b = math.exp(lb)
x = (x_in - u) / s
Ea = (a / 2 + x)
Eb = (b / 2 - x)
Ea[Ea < -25] = -25
Eb[Eb < -25] = -25
return A * a * b / (a + b) * np.exp(-x**2) * (
erfcx(Eb) + erfcx(Ea)) / 1.1273434771994788
def __call__(self, w):
sigma = self.p[3]
x0 = self.p[1]
g = self.p[2]
a = sqrt(w**2 - 1j*g*w)
a = a.real - 1j*a.imag #we want the imaginary part to the root to be positive
prefac = 1j*sqrt(3.14149)*self.p[0]*x0**2/(sqrt(22)*sigma)
eps = prefac*exp(-.5)*(1/a)*( sp.erfcx(-1j*(a-x0)/sigma) + sp.erfcx(-1j*(a+x0)/sigma) )
#self.f = 2*prefac*exp(-x0**2/(2*sigma**2))*(1 + sp.erf(1j*x0/sigma))
self.f = eps.real[1]
return (eps.real, eps.imag)
def func_dawson(self, x: Tensor) -> Tensor:
if x.is_cuda:
if x.numel() > mnn_config.get_value('cpu_or_gpu'):
return self._gpu_dawson(x)
else:
device = x.device
return torch.from_numpy(
scipy.erfcx(-x.cpu().numpy()) * math.sqrt(math.pi) /
2).to(device=device)
else:
return torch.from_numpy(
scipy.erfcx(-x.numpy()) * math.sqrt(math.pi) / 2)
def __call__(self, w):
sigma = self.p[3]
x0 = self.p[1]
g = self.p[2]
a = sqrt(w**2 - 1j * g * w)
a = a.real - 1j * a.imag #we want the imaginary part to the root to be positive
prefac = 1j * sqrt(3.14149) * self.p[0] * x0**2 / (sqrt(22) * sigma)
eps = prefac * exp(-.5) * (1 / a) * (sp.erfcx(-1j * (a - x0) / sigma) +
sp.erfcx(-1j * (a + x0) / sigma))
#self.f = 2*prefac*exp(-x0**2/(2*sigma**2))*(1 + sp.erf(1j*x0/sigma))
self.f = eps.real[1]
return (eps.real, eps.imag)
def int_exact(self, X):
q = np.zeros(X.size)
i = 0
fun1 = lambda x: np.power(erfcx(-x), 2) * dawsn(x)
fun2 = lambda x: np.exp(-x * x) * np.power(erfcx(-x), 2)
for x in X:
y1, _ = quad(fun1, -np.inf, x)
y2, _ = quad(fun2, -np.inf, x)
q[i] = -np.pi / 4 * y1 + np.power(np.sqrt(np.pi) / 2,
3) * erfi(x) * y2
i += 1
return q
def test():
relative_error = 0
for i in range(100):
x = -1 + i * (10 - (-1)) / 100
my_erfcx = erfcx(torch.FloatTensor([x]))
relative_error = relative_error + np.abs(
my_erfcx.item() - special.erfcx(x)) / special.erfcx(x)
average_error = relative_error / 100
print(average_error)
normal = Normal(loc=torch.Tensor([0.0]), scale=torch.Tensor([1.0]))
# cdf from 0 to x
print(normal.cdf(1.6449))
print(normal.icdf(torch.Tensor([0.95])))
def int_exact(self, X):
'''Integral of the 2nd order Dawson function (with a change of order of integration)'''
q = np.zeros(X.size)
i = 0
fun1 = lambda x: np.power(erfcx(-x), 2) * dawsn(x)
fun2 = lambda x: np.exp(-x * x) * np.power(erfcx(-x), 2)
for x in X:
if x < -25: #== -np.inf:
q[i] = self.int_asym_neginf(x)
else:
y1, _ = quad(fun1, -np.inf, x)
y2, _ = quad(fun2, -np.inf, x)
q[i] = -np.pi / 4 * y1 + np.power(np.sqrt(np.pi) / 2,
3) * erfi(x) * y2
i += 1
return q
def mushroom_math(x, delta=.1, vf=.1):
"""
new method, rewrite for numerical stability at high delta
delta=0 causes divide by zero error!! Compensate elsewhere.
http://ab-initio.mit.edu/wiki/index.php/Faddeeva_Package
"""
from scipy.special import erfc, erfcx
x_half = x / 2.0
delta_double = 2.0 * delta
return (
(erfc(x_half) - erfcx(delta_double + x_half) / exp(x_half * x_half) -
erfc(x) +
((.25 - delta *
(x + delta_double)) * erfcx(delta_double + x) + delta / SQRT_PI) *
4.0 / exp(x * x)) * vf / (delta_double * erfcx(delta_double)))
def kai(model, altmp): #approximated 1/Psi
M = model.M
kaitmp = np.where(altmp / sqrt(2.0) < -10.0, -altmp, np.zeros((M)))
kaitmp2 = np.where(
abs(altmp / sqrt(2.0)) < 10.0,
sqrt(2.0 / pi) / scisp.erfcx(-altmp / sqrt(2.0)), kaitmp)
return kaitmp2
def _erfcx_integral(a, b, order):
"""Fixed order Gauss-Legendre quadrature of erfcx from a to b."""
assert np.all(a >= 0) and np.all(b >= 0)
x, w = roots_legendre(order)
x = x[:, np.newaxis]
w = w[:, np.newaxis]
return (b - a) * np.sum(w * erfcx((b - a) * x / 2 +
(b + a) / 2), axis=0) / 2
def score_sum(sum_u, k, l, data):
m, n = data.shape
cnr = gammaln(m + 1) - gammaln(k + 1) - gammaln(m - k + 1)
cnc = gammaln(n + 1) - gammaln(l + 1) - gammaln(n - l + 1)
ar = sum_u / np.sqrt(k * l)
rest2 = -(ar * ar) / 2.0 + np.log(erfcx(ar / np.sqrt(2)) * 0.5)
sc = -rest2 - cnr - cnc
return sc
def log_erfc(x):
fx = np.zeros(x.shape)
ind = (x > 8)
fx[ind] = np.log(erfcx(x[ind])) - x[ind]**2
ind = (x <= 8)
fx[ind] = np.log(erfc(x[ind]))
return fx
def cvoigt(omega, cent, delta, gamma=0.0):
"""Complex normalized Voigt line shape
"""
a = (delta**2) / (4.0 * numpy.log(2))
z = (gamma - 1j * (omega - cent)) / (2.0 * numpy.sqrt(a))
return numpy.real(special.erfcx(z)) * numpy.sqrt(numpy.pi / a) / 2.0
def channel(y, w, v, var_noise):
"""Compute g and g' for probit channel"""
phi = -y * w / np.sqrt(2 * (v + var_noise))
g = 2 * y / (np.sqrt(2 * np.pi * (v + var_noise)) * erfcx(phi))
dg = -g * (w / (v + var_noise) + g)
return g, dg
def forward_cpu(self, x):
if not available_cpu:
raise ImportError('SciPy is not available. Forward computation'
' of erfcx in CPU cannot be done. ' +
str(_import_error))
self.retain_inputs((0,))
self.retain_outputs((0,))
return utils.force_array(special.erfcx(x[0]), dtype=x[0].dtype),
def forward_cpu(self, x):
if not available_cpu:
raise ImportError('SciPy is not available. Forward computation'
' of erfcx in CPU cannot be done. ' +
str(_import_error))
self.retain_inputs((0, ))
self.retain_outputs((0, ))
return utils.force_array(special.erfcx(x[0]), dtype=x[0].dtype),
def cvoigt(omega, cent, delta, gamma=0.0):
"""Complex normalized Voigt line shape
"""
a = (delta**2)/(4.0*numpy.log(2))
z = (gamma - 1j*(omega - cent))/(2.0*numpy.sqrt(a))
return numpy.real(special.erfcx(z))*numpy.sqrt(numpy.pi/a)/2.0
def lnDifErf(z1, z2):
#Z2 is always positive
logdiferf = np.zeros(z1.shape)
ind = np.where(z1 > 0.)
ind2 = np.where(z1 <= 0.)
if ind[0].shape > 0:
z1i = z1[ind]
z12 = z1i * z1i
z2i = z2[ind]
logdiferf[ind] = -z12 + np.log(
erfcx(z1i) - erfcx(z2i) * np.exp(z12 - z2i**2))
if ind2[0].shape > 0:
z1i = z1[ind2]
z2i = z2[ind2]
logdiferf[ind2] = np.log(erf(z2i) - erf(z1i))
return logdiferf
def testLogErfcx(self, dtype):
x = tf.random.uniform(shape=[int(1e5)],
minval=-3.,
maxval=3.,
dtype=dtype,
seed=test_util.test_seed())
x_, logerfcx_ = self.evaluate([x, tfp.math.logerfcx(x)])
self.assertAllClose(np.log(scipy_special.erfcx(x_)), logerfcx_)
def testErfcxLargeNegative(self, dtype):
x = tf.random.uniform(shape=[int(1e5)],
minval=-100.,
maxval=-20.,
dtype=dtype,
seed=test_util.test_seed())
x_, erfcx_ = self.evaluate([x, tfp.math.erfcx(x)])
self.assertAllClose(scipy_special.erfcx(x_), erfcx_)
def int_brute_force(self, X):
'''2nd order Dawson function (direct integration)'''
q = np.zeros(X.size)
i = 0
for x in X:
q[i], _ = quad(lambda x: erfcx(-x), 0, x)
i += 1
q = q * np.sqrt(np.pi) / 2
return q
def dbl_dawson(x):
assert isinstance(x, np.ndarray)
# assert x.ndim == 1
length = x.size
shape = x.shape
x_flatten = x.flatten()
res = np.empty([length], dtype=np.float)
for i in range(length):
res[i] = pi / 4 * quad(lambda u: np.exp(x_flatten[i]**2 - u**2) * erfcx(-u) ** 2, - np.inf, x_flatten[i])[0]
return res.reshape(shape)
def g_prec(I,theta,sigm):
range_grid = np.meshgrid(np.linspace(0,1,nint),xrange(len(I)))[0]
dint = (theta-V_r)/(sigm*nint)
#int_mesh = np.ndarray((480,nint))
diff_grid = np.meshgrid(xrange(nint),(theta-V_r)/sigm)[1]
low_value_grid = np.meshgrid(xrange(nint),(V_r-E_l-I*tau_m)/sigm)[1]
int_mesh = erfcx(-(low_value_grid + range_grid*diff_grid))
return 1./(pi_sqrt_tau*dint*int_mesh.sum(axis=1))
def compute_F( t, x_bar, h ):
if x_bar < 1e-30:
return math.sqrt(h)*sp.erf( np.pi/(2*h*t) )
else:
z = np.complex( np.pi/(2*h*t), t*x_bar )
w_iz = sp.erfcx( z )
f = math.exp( -t**2 * x_bar**2 )
f = f - np.real( np.exp(-t**2 * x_bar**2 - z*z)*w_iz )
f = f*math.sqrt(h)
return f
def phi_noise(I,V_t,V_r,E_l,tau_m,sigm,nint): range_grid = np.meshgrid(np.linspace(0,1,nint),xrange(len(I)))[0] dint = (V_t-V_r)/(sigm*nint) #int_mesh = np.ndarray((480,nint)) diff_grid = np.meshgrid(xrange(nint),(V_t-V_r)/sigm)[1] low_value_grid = np.meshgrid(xrange(nint),(V_r-E_l-I*tau_m)/sigm)[1] int_mesh = erfcx(-(low_value_grid + range_grid*diff_grid)) return 1./(np.sqrt(np.pi)*tau_m*dint*int_mesh.sum(axis=1))
def compute_F( t, x_bar, h ):
if x_bar < 1e-30:
return sqrt(h)*sp.erf( pi/(2*h*t) )
else:
z = np.complex( pi/(2*h*t), t*x_bar )
w_iz = sp.erfcx( z )
f = exp( -t**2 * x_bar**2 )
f = f - np.real( np.exp(-t**2 * x_bar**2 - z*z)*w_iz )
f = f*sqrt(h)
return f
def integrand(u_arr):
"""Integrand of self-consistency equation"""
integrand_all = erfcx(-u_arr)
#integrand_all = np.zeros(u_arr.shape)
#u_mask = u_arr < -4.0
#u = u_arr[u_mask]
#integrand_all[u_mask] = -1. / np.sqrt(np.pi) * (1.0 / u - 1.0 / (2.0 * u**3) +
#3.0 / (4.0 * u**5) -
#15.0 / (8.0 * u**7))
#integrand_all[~u_mask] = np.exp(u_arr[~u_mask]**2) * (1. + erf(u_arr[~u_mask]))
return integrand_all
def compute_expectation_spec(beta, mu, sigma):
"""compute \int(exp(-beta*x)*phi(x;mu,sigma^2)) from mu to infty"""
print beta, mu, sigma
# we do this because the first term is liable to overflow
u = beta*sigma/sqrt(2)
prefactor = 1/2.0 * exp(-beta*mu)
# first_term_ref = exp(u**2)
# second_term_ref = erfc(u)
# ans_ref = prefactor * first_term * second_term
ans = prefactor * erfcx(u)
# print ans_ref, ans
return ans
def mushroom_math(x, delta=0.1, vf=0.1):
"""
new method, rewrite for numerical stability at high delta
delta=0 causes divide by zero error!! Compensate elsewhere.
http://ab-initio.mit.edu/wiki/index.php/Faddeeva_Package
"""
from scipy.special import erfc, erfcx
x_half = x / 2.0
delta_double = 2.0 * delta
return (
(
erfc(x_half)
- erfcx(delta_double + x_half) / exp(x_half * x_half)
- erfc(x)
+ ((0.25 - delta * (x + delta_double)) * erfcx(delta_double + x) + delta / SQRT_PI) * 4.0 / exp(x * x)
)
* vf
/ (delta_double * erfcx(delta_double))
)
def compute_expectation_spec(beta, mu, sigma):
"""compute \int(exp(-beta*x)*phi(x;mu,sigma^2)) from mu to infty"""
print beta, mu, sigma
# we do this because the first term is liable to overflow
u = beta * sigma / sqrt(2)
prefactor = 1 / 2.0 * exp(-beta * mu)
# first_term_ref = exp(u**2)
# second_term_ref = erfc(u)
# ans_ref = prefactor * first_term * second_term
ans = prefactor * erfcx(u)
# print ans_ref, ans
return ans
def CreateData(y_range=(0., 2.), npoints=1000000, filename='erfcxinv.h5'):
# Create a table of values for erfc(x), so that we can numerically estimate the inverse.
# We're only interested in a limited range of values.
x = np.flip(
np.linspace(*y_range, npoints)
) # function monotonically decreases, we want f(x) to increase with index, thus perform flip
y = spec.erfcx(x)
result = np.column_stack((y, x))
f = h5.File(filename, 'w')
dset = f.create_dataset('erfcxinv',
data=result,
compression='gzip',
compression_opts=7)
f.close()
def sigmoid_lif(i0,
tau=1.0,
sigma=1.0,
vr=-68.0,
vth=-48.0,
vrevers=-50.0,
dt=10E-3):
x1 = (vr - vrevers - i0) / sigma
x2 = (vth - vrevers - i0) / sigma
dx = (x2 - x1) / 2000.0
x = np.arange(x1, x2 + dx, dx, dtype=np.float64)
fx = erfcx(-x)
t = tau * np.sqrt(pi) * np.sum(fx * dx) + dt * tau
return 1.0 / t
def test_erfc(self):
# in the application in STO-GTO integration,
# the argument of erfcx becomes a/(2.0*sqrt(b)),
# where a and b is orbital exponent of STO and GTO.
# a ~ 1.0, 0.5 or 1/3
# b ~ 0.01 ~ 30.0
# So argument becomes
# 5 ~ 0.027
for theta in [0, 1, 45, 90, 270]:
t = theta * np.pi / 180.0
for a in [0.001, 0.01, 0.1, 1.0, 10.0, 100.0]:
z = a * np.exp(1.0j * a)
self.assertAlmostEqual(sp.erfc(z), erfc(z), places=15)
self.assertAlmostEqual(sp.erfcx(z), erfcx(z), places=15)
def xGaussFn(mode, x, h, mu, sig, tau):
if mode == 1:
return (h * sig * np.sqrt(np.pi / 2.) /
tau) * np.exp((sig / tau)**2. / 2. - (x - mu) / tau) * sp.erfc(
(sig / tau - (x - mu) / sig) / np.sqrt(2.))
elif mode == 2:
return (h * sig * np.sqrt(np.pi / 2.) / tau) * np.exp(
(-1. / 2.) * ((x - mu) / sig)**2.) * sp.erfcx(
(sig / tau - (x - mu) / sig) / np.sqrt(2.))
elif mode == 3:
return (h / (1 - (x - mu) * tau / sig**2.)) * np.exp(
(-1. / 2.) * ((x - mu) / sig)**2.)
else:
print "unknown mode!"
return -1
def test_erfc(self):
# in the application in STO-GTO integration,
# the argument of erfcx becomes a/(2.0*sqrt(b)),
# where a and b is orbital exponent of STO and GTO.
# a ~ 1.0, 0.5 or 1/3
# b ~ 0.01 ~ 30.0
# So argument becomes
# 5 ~ 0.027
for theta in [0, 1, 45, 90, 270]:
t = theta * np.pi / 180.0
for a in [0.001, 0.01, 0.1, 1.0, 10.0, 100.0]:
z = a * np.exp(1.0j * a)
self.assertAlmostEqual(sp.erfc(z), erfc(z), places=15)
self.assertAlmostEqual(sp.erfcx(z), erfcx(z), places=15)
def analyticDistribution(x, args):
#args[0]= time args[1]=absorption coefficient
t = args[0]
alpha = args[1]
mat = args[2]
z = mat.D * t
x1 = alpha * np.sqrt(z) - x / (2 * np.sqrt(z))
x2 = alpha * np.sqrt(z) + x / (2 * np.sqrt(z))
x3 = mat.s * np.sqrt(t / mat.D) + x / 2. / np.sqrt(z)
phi=np.array(0.5*(erfcx(x1)+\
(alpha*mat.D+mat.s)/(alpha*mat.D-mat.s)*erfcx(x2))\
-mat.s*erfcx(x3)/(alpha*mat.D-mat.s))
phi[phi == inf] = 0
np.seterr(under='ignore')
dat = phi * np.exp(-x * x / (4 * z) - t / mat.tau)
np.seterr(under='raise')
dat[np.where(dat == 0)] = np.exp(-alpha * x[np.where(dat == 0)])
return dat
def trunc_norm_param_residuals(params):
mean = params[:d]
chol = params[d:]
alpha = -mean * chol
erf_term = 1 / erfcx(alpha / 2**0.5)
std = 1 / chol
var = std**2
const = (2 / np.pi)**0.5
trunc_mean = mean + const * erf_term * std
trunc_var = var * (1 + const * alpha * erf_term - (const * erf_term)**2)
mean_res = sample_mean - trunc_mean # (d, )
var_res = sample_var - trunc_var # (d, )
return np.concatenate((mean_res, var_res))
def _npcr(self, x):
"""
Return the pdf/cdf ratio of a standard normal RV evaluated at x.
Parameters
----------
x : ndarray
The values to evaluate the ratio at.
Returns
-------
npcr_values : ndarray
The evaluated ratios.
"""
convert = x.dtype.type
npcr_values = (2 / np.sqrt(2 * np.pi)) / convert(
special.erfcx(-np.float_(x / np.sqrt(2))))
return npcr_values
def ln_diff_erfs(x1, x2, return_sign=False):
"""Function for stably computing the log of difference of two erfs in a numerically stable manner.
:param x1 : argument of the positive erf
:type x1: ndarray
:param x2 : argument of the negative erf
:type x2: ndarray
:return: tuple containing (log(abs(erf(x1) - erf(x2))), sign(erf(x1) - erf(x2)))
Based on MATLAB code that was written by Antti Honkela and modified by David Luengo and originally derived from code by Neil Lawrence.
"""
x1 = np.require(x1).real
x2 = np.require(x2).real
if x1.size==1:
x1 = np.reshape(x1, (1, 1))
if x2.size==1:
x2 = np.reshape(x2, (1, 1))
if x1.shape==x2.shape:
v = np.zeros_like(x1)
else:
if x1.size==1:
v = np.zeros(x2.shape)
elif x2.size==1:
v = np.zeros(x1.shape)
else:
raise ValueError("This function does not broadcast unless provided with a scalar.")
if x1.size == 1:
x1 = np.tile(x1, x2.shape)
if x2.size == 1:
x2 = np.tile(x2, x1.shape)
sign = np.sign(x1 - x2)
if x1.size == 1:
if sign== -1:
swap = x1
x1 = x2
x2 = swap
else:
I = sign == -1
swap = x1[I]
x1[I] = x2[I]
x2[I] = swap
with np.errstate(divide='ignore'):
# switch off log of zero warnings.
# Case 0: arguments of different sign, no problems with loss of accuracy
I0 = np.logical_or(np.logical_and(x1>0, x2<0), np.logical_and(x2>0, x1<0)) # I1=(x1*x2)<0
# Case 1: x1 = x2 so we have log of zero.
I1 = (x1 == x2)
# Case 2: Both arguments are non-negative
I2 = np.logical_and(x1 > 0, np.logical_and(np.logical_not(I0),
np.logical_not(I1)))
# Case 3: Both arguments are non-positive
I3 = np.logical_and(np.logical_and(np.logical_not(I0),
np.logical_not(I1)),
np.logical_not(I2))
_x2 = x2.flatten()
_x1 = x1.flatten()
for group, flags in zip((0, 1, 2, 3), (I0, I1, I2, I3)):
if np.any(flags):
if not x1.size==1:
_x1 = x1[flags]
if not x2.size==1:
_x2 = x2[flags]
if group==0:
v[flags] = np.log( erf(_x1) - erf(_x2) )
elif group==1:
v[flags] = -np.inf
elif group==2:
v[flags] = np.log(erfcx(_x2)
-erfcx(_x1)*np.exp(_x2**2
-_x1**2)) - _x2**2
elif group==3:
v[flags] = np.log(erfcx(-_x1)
-erfcx(-_x2)*np.exp(_x1**2
-_x2**2))-_x1**2
# TODO: switch back on log of zero warnings.
if return_sign:
return v, sign
else:
if v.size==1:
if sign==-1:
v = v.view('complex64')
v += np.pi*1j
else:
# Need to add in a complex part because argument is negative.
v = v.view('complex64')
v[I] += np.pi*1j
return v
def differfln(x0, x1):
# this is a, hopefully!, a numerically more stable variant of log(erf(x0)-erf(x1)) = log(erfc(x1)-erfc(x0)).
return np.where(x0>x1, -x1*x1 + np.log(erfcx(x1)-np.exp(-x0**2+x1**2)*erfcx(x0)), -x0*x0 + np.log(np.exp(-x1**2+x0**2)*erfcx(x1) - erfcx(x0)))
def _erfcx_cpu(x, dtype):
from scipy import special
return special.erfcx(x).astype(dtype)
def normcdfln(x):
return np.where(x < 0, -.5*x*x + np.log(.5) + np.log(erfcx(-x/np.sqrt(2))), np.log(normcdf(x)))
def integrand(u):
"""Integrand of self-consistency equation"""
return erfcx(-u)
