def _get_I_total(R0, d, t):
A = np.ln(R0)
B = np.ln(1+ d)
mu = 0.5 * A / B
I = 0.5 * np.exp( 0.25 * A**2 / B * np.sqrt( np.pi / B))
I = I * np.sqrt(B) * (np.erf(t - mu)- np.erf(-mu))
return I
def _get_I_total(R0, d, t):
A = np.ln(R0)
B = np.ln(1 + d)
mu = 0.5 * A / B
I = 0.5 * np.exp(0.25 * A**2 / B * np.sqrt(np.pi / B))
I = I * np.sqrt(B) * (np.erf(t - mu) - np.erf(-mu))
return I
def _black_scholes(self, nopt, price, strike, t, rate, vol):
mr = -rate
sig_sig_two = vol * vol * 2
P = price
S = strike
T = t
a = log(P / S)
b = T * mr
z = T * sig_sig_two
c = 0.25 * z
y = invsqrt(z)
w1 = (a - b + c) * y
w2 = (a - b - c) * y
d1 = 0.5 + 0.5 * erf(w1)
d2 = 0.5 + 0.5 * erf(w2)
Se = exp(b) * S
call = P * d1 - Se * d2
put = call - P + Se
return (call, put)
def __init__(self,
func=lambda x: np.sin(x),
freq=440,
vol=1.0,
random=False,
length=def_length):
(pb_freq, pb_bits, pb_chns) = pg.mixer.get_init()
self.freq = freq
self.multiplier = int(freq * length)
#self.multiplier = freq * length
self.length = max(1, int(float(pb_freq) / freq * self.multiplier))
self.lin = np.linspace(0.0,
self.multiplier,
self.length,
endpoint=False)
#self.lin2 = np.linspace(0.0,m
#self.lin = np.linspace(0.0, )
#self.ary = np.sin(self.lin * 2.0 * np.pi)
self.ary = func(self.lin * 2 * np.pi)
fade_fac = 1.
fadeout_len = min(50, int(len(self.lin) * 0.01))
self.fadein = np.arctan(self.lin / fade_fac) / (np.pi / 2.)
#self.fadeout= (1. + np.arctan(- ( self.lin - self.lin[-fadeout_len])/fade_fac )/(np.pi/2.))
self.fadein = np.erf(self.lin)
self.fadeout = 0.5 - np.erf(self.lin - self.lin[-50]).astype('f2') / 2.
self.fadeinout = np.multiply(self.fadein, self.fadeout)
self.ary = np.multiply(self.fadeinout, self.ary)
norm = abs(1. / self.ary.max())
self.ary = self.ary * norm
norm2 = abs(1. / self.ary.min())
self.ary = self.ary * norm2
# If mixer is in stereo mode, double up the array information for
# each channel.
if pb_chns == 2:
self.ary2 = np.repeat(self.ary[..., np.newaxis], 2, axis=1)
else:
self.ary2 = self.ary
if pb_bits == 8:
snd_ary = self.ary2 * vol * 127.0
self.sound = pg.sndarray.make_sound(snd_ary.astype(np.uint8) + 128)
elif pb_bits == -16:
snd_ary = self.ary2 * vol * float((1 << 15) - 1)
self.sound = pg.sndarray.make_sound(snd_ary.astype(np.int16))
else:
print "Sound playback resolution unsupported (either 8 or -16)."
return None
def __init__(self, func = lambda x: np.sin(x ) ,freq=440 , vol=1.0, random=False,
length=def_length ):
(pb_freq, pb_bits, pb_chns) = pg.mixer.get_init()
self.freq=freq
self.multiplier = int(freq * length)
#self.multiplier = freq * length
self.length = max(1, int(float(pb_freq) / freq * self.multiplier))
self.lin = np.linspace(0.0, self.multiplier, self.length, endpoint=False)
#self.lin2 = np.linspace(0.0,m
#self.lin = np.linspace(0.0, )
#self.ary = np.sin(self.lin * 2.0 * np.pi)
self.ary = func(self.lin*2*np.pi)
fade_fac=1.
fadeout_len = min(50, int(len(self.lin)*0.01 ))
self.fadein= np.arctan(self.lin/fade_fac)/(np.pi/2.)
#self.fadeout= (1. + np.arctan(- ( self.lin - self.lin[-fadeout_len])/fade_fac )/(np.pi/2.))
self.fadein = np.erf( self.lin )
self.fadeout = 0.5 - np.erf(self.lin- self.lin[-50]).astype('f2')/2.
self.fadeinout = np.multiply(self.fadein, self.fadeout)
self.ary = np.multiply(self.fadeinout,self.ary)
norm = abs(1./self.ary.max())
self.ary = self.ary* norm
norm2 = abs(1./self.ary.min())
self.ary = self.ary*norm2
# If mixer is in stereo mode, double up the array information for
# each channel.
if pb_chns == 2:
self.ary2 = np.repeat(self.ary[..., np.newaxis], 2, axis=1)
else:
self.ary2 = self.ary
if pb_bits == 8:
snd_ary = self.ary2 * vol * 127.0
self.sound= pg.sndarray.make_sound(snd_ary.astype(np.uint8) + 128)
elif pb_bits == -16:
snd_ary = self.ary2 * vol * float((1 << 15) - 1)
self.sound= pg.sndarray.make_sound(snd_ary.astype(np.int16))
else:
print "Sound playback resolution unsupported (either 8 or -16)."
return None
def _broaden_wmp_polynomials(E, dopp, n):
r"""Evaluate Doppler-broadened windowed multipole curvefit.
The curvefit is a polynomial of the form :math:`\frac{a}{E}
+ \frac{b}{\sqrt{E}} + c + d \sqrt{E} + \ldots`
Parameters
----------
E : Real
Energy to evaluate at.
dopp : Real
sqrt(atomic weight ratio / kT) in units of eV.
n : Integral
Number of components to the polynomial.
Returns
-------
numpy.ndarray
The value of each Doppler-broadened curvefit polynomial term.
"""
sqrtE = np.sqrt(E)
beta = sqrtE * dopp
half_inv_dopp2 = 0.5 / dopp**2
quarter_inv_dopp4 = half_inv_dopp2**2
if beta > 6.0:
# Save time, ERF(6) is 1 to machine precision.
# beta/sqrtpi*exp(-beta**2) is also approximately 1 machine epsilon.
erf_beta = 1.0
exp_m_beta2 = 0.0
else:
erf_beta = np.erf(beta)
exp_m_beta2 = np.exp(-beta**2)
# Assume that, for sure, we'll use a second order (1/E, 1/V, const)
# fit, and no less.
factors = np.zeros(n)
factors[0] = erf_beta / E
factors[1] = 1.0 / sqrtE
factors[2] = (factors[0] * (half_inv_dopp2 + E) + exp_m_beta2 /
(beta * np.sqrt(np.pi)))
# Perform recursive broadening of high order components. range(1, n-4)
# replaces a do i = 1, n=3. All indices are reduced by one due to the
# 1-based vs. 0-based indexing.
for i in range(1, n - 4):
if i != 1:
factors[i +
2] = (-factors[i - 2] * (i - 1.0) * i * quarter_inv_dopp4 +
factors[i] * (E + (1.0 + 2.0 * i) * half_inv_dopp2))
else:
# Although it's mathematically identical, factors[0] will contain
# nothing, and we don't want to have to worry about memory.
factors[i + 2] = factors[i] * (E +
(1.0 + 2.0 * i) * half_inv_dopp2)
return factors
def register_element_ops():
binary_op_list = [
["mul", lambda a, b: a * b],
["add", lambda a, b: a + b],
["sub", lambda a, b: a - b],
["div", lambda a, b: a / (b + 1e-4)],
[
"pow",
lambda a, b: torch.pow(a, b),
lambda a, b: np.power(a, b),
], # no fuson triggered
["max", lambda a, b: torch.max(a, b), lambda a, b: np.maximum(a, b)],
["min", lambda a, b: torch.min(a, b), lambda a, b: np.minimum(a, b)],
]
unary_op_list = [
["erf", lambda x: torch.erf(x), lambda x: np.erf(x)],
["exp", lambda x: torch.exp(x), lambda x: np.exp(x)],
["sin", lambda x: torch.sin(x), lambda x: np.sin(x)],
["cos", lambda x: torch.cos(x), lambda x: np.cos(x)],
[
"rand_like", lambda x: torch.rand_like(x),
lambda x: np.random.rand(*x.shape)
],
]
for split_input, binary_op in itertools.product([True, False],
binary_op_list):
# Make a copy of ElementBench
if len(binary_op) == 2:
[op_str, op_pt_func] = binary_op
op_np_func = op_pt_func
elif len(binary_op) == 3:
[op_str, op_pt_func, op_np_func] = binary_op
split_str = "split" if split_input else "shared"
op_str = split_str + "_" + op_str
bm_cls = type("ElementBench_" + op_str, (ElementBench, ), {})
bm_cls.op_str = op_str
bm_cls.binary_op_pt_func = op_pt_func
bm_cls.binary_op_np_func = op_np_func
bm_cls.split_input = split_input
benchmark.register_benchmark_class(bm_cls)
for split_input, unary_op in itertools.product([True, False],
unary_op_list):
# Make a copy of ElementBench
if len(unary_op) == 2:
[op_str, op_pt_func] = unary_op
op_np_func = op_pt_func
elif len(unary_op) == 3:
[op_str, op_pt_func, op_np_func] = unary_op
split_str = "split" if split_input else "shared"
op_str = split_str + "_" + op_str
bm_cls = type("ElementBench_" + op_str, (ElementBench, ), {})
bm_cls.op_str = op_str
bm_cls.unary_op_pt_func = op_pt_func
bm_cls.unary_op_np_func = op_np_func
bm_cls.split_input = split_input
benchmark.register_benchmark_class(bm_cls)
def int_local_time(self, t, K):
"""Explicit form for the integral of the local time between 0 and t
"""
dJ = self.dJ(0, K)
J = self.J(0, K)
return dJ * np.exp(-J**2 / (2 * t)) * np.sqrt(
2 / np.pi) * np.sqrt(t) + dJ * J * (np.erf(J /
(np.sqrt(2 * t))) - 1)
def _broaden_wmp_polynomials(E, dopp, n):
r"""Evaluate Doppler-broadened windowed multipole curvefit.
The curvefit is a polynomial of the form :math:`\frac{a}{E}
+ \frac{b}{\sqrt{E}} + c + d \sqrt{E} + \ldots`
Parameters
----------
E : Real
Energy to evaluate at.
dopp : Real
sqrt(atomic weight ratio / kT) in units of eV.
n : Integral
Number of components to the polynomial.
Returns
-------
numpy.ndarray
The value of each Doppler-broadened curvefit polynomial term.
"""
sqrtE = np.sqrt(E)
beta = sqrtE * dopp
half_inv_dopp2 = 0.5 / dopp**2
quarter_inv_dopp4 = half_inv_dopp2**2
if beta > 6.0:
# Save time, ERF(6) is 1 to machine precision.
# beta/sqrtpi*exp(-beta**2) is also approximately 1 machine epsilon.
erf_beta = 1.0
exp_m_beta2 = 0.0
else:
erf_beta = np.erf(beta)
exp_m_beta2 = np.exp(-beta**2)
# Assume that, for sure, we'll use a second order (1/E, 1/V, const)
# fit, and no less.
factors = np.zeros(n)
factors[0] = erf_beta / E
factors[1] = 1.0 / sqrtE
factors[2] = (factors[0] * (half_inv_dopp2 + E)
+ exp_m_beta2 / (beta * np.sqrt(np.pi)))
# Perform recursive broadening of high order components. range(1, n-4)
# replaces a do i = 1, n=3. All indices are reduced by one due to the
# 1-based vs. 0-based indexing.
for i in range(1, n-4):
if i != 1:
factors[i+2] = (-factors[i-2] * (i - 1.0) * i * quarter_inv_dopp4
+ factors[i] * (E + (1.0 + 2.0 * i) * half_inv_dopp2))
else:
# Although it's mathematically identical, factors[0] will contain
# nothing, and we don't want to have to worry about memory.
factors[i+2] = factors[i]*(E + (1.0 + 2.0 * i) * half_inv_dopp2)
return factors
def erf(f=Ellipsis):
'''
erf(x) yields a potential function that calculates the error function over the input x. If x is
a constant, yields a constant potential function.
erf() is equivalent to erf(...), which is just the error function, calculated over its inputs.
'''
f = to_potential(f)
if is_const_potential(f): return const_potential(np.erf(f.c))
elif is_identity_potential(f): return ErfPotential()
else: return compose(ErfPotential(), f)
def expected_improvement(self, fref, f, s):
if s > 0.0:
if self.MIN[self.nfg]:
y = (fref - f)/s
else:
y = (f - fref)/s
cdf = 0.5*(1.0 - np.erf(-y/np.sqrt(2.0)))
pdf = 1.0/np.sqrt(2.0*np.pi)*np.exp(-0.5*y**2.0)
ei = s*(y*cdf + pdf)
else:
ei = 0.0
return ei
def grid_coherence(U, res, f0, model='Pierson-Moskowitz'):
""" Calculates the coherence time of a resolution cell
:param U: wind velocity at reference height
:param res: grid size [m]
:param f0: carrier frequency [Hz]
:param model: Model used, defaults to Pierson-Moskowitz (currently
the only one implemented)
"""
tau_c = 3.29 * const.c / f0 / U / np.sqrt(np.erf(2.688 * res / U**2))
return tau_c
def whrankHamm(q_codes, db_codes, q_feats, fmu, fstd, w_type='ones'):
if w_type == 'ones':
weights = np.ones_like(q_feats)
elif w_type == 'q':
weights = np.abs(q_feats)
elif w_type == 'std':
weights = np.ones_like(q_feats) / fstd
elif w_type == 'q_std':
weights = np.abs(q_feats) / fstd
elif w_type == 'erf':
Pr = 0.5 * (1 + q_codes * np.erf((-q_feats-fmu) / (np.sqrt(2)*fstd)))
weights = np.log((1 - Pr) / Pr)
num1 = q_codes.shape[0]
num2 = db_codes.shape[0]
distMat = np.zeros((num1, num2))
for i in range(num1):
codediff = np.abs(np.tile(q_codes[i], (num2, 1)) - db_codes) / 2
distMat[i] = np.dot(weights[i], codediff.transpose())
return distMat
def __init__(self, alpha=2, Mach=10, epsilon=0.2, b=0.4,
phi_t=1/3., alpha_vir=1., Beta=20,
):
self.alpha = alpha
self.b = b
self.epsilon = epsilon
self.Mach = Mach
self.alpha_vir = alpha_vir
self.phi_t = phi_t
self.Beta = Beta
# eqn19
self.s_t = 0.5 * (2*np.abs(self.alpha) - 1) * self.sigma_s**2
self.C = (np.exp(0.5 * (self.alpha-1)
* self.alpha * self.sigma_s**2)
/ (self.sigma_s * np.sqrt(2*np.pi)))
self.N = 2 * (1 + np.erf((2*np.log(self.s_t) +
self.sigma_s**2)/(2**1.5*self.sigma_s))
+ (2*self.C * np.exp(-self.s_t*self.alpha))/self.alpha)**-1
def value(self, x):
return np.erf(flattest(x))
def test_unary_fun(self):
import math
data = [1., 2., 3.]
c = np.array(data)
d = np.acos(c / 3)
for i, ei in enumerate(d):
self.assertEqual(ei, math.acos(data[i] / 3))
d = np.asin(c / 3)
for i, ei in enumerate(d):
self.assertEqual(ei, math.asin(data[i] / 3))
d = np.atan(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.atan(data[i]))
d = np.sin(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.sin(data[i]))
d = np.cos(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.cos(data[i]))
d = np.tan(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.tan(data[i]))
d = np.acosh(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.acosh(data[i]))
d = np.asinh(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.asinh(data[i]))
d = np.atanh(c / 3.1)
for i, ei in enumerate(d):
self.assertEqual(ei, math.atanh(data[i] / 3.1))
d = np.sinh(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.sinh(data[i]))
d = np.cosh(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.cosh(data[i]))
d = np.tanh(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.tanh(data[i]))
d = np.ceil(c * 2.7)
for i, ei in enumerate(d):
self.assertEqual(ei, math.ceil(data[i] * 2.7))
d = np.floor(c * 2.7)
for i, ei in enumerate(d):
self.assertEqual(ei, math.floor(data[i] * 2.7))
d = np.erf(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.erf(data[i]))
d = np.erfc(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.erfc(data[i]))
d = np.exp(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.exp(data[i]))
d = np.expm1(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.expm1(data[i]))
d = np.gamma(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.gamma(data[i]))
d = np.lgamma(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.lgamma(data[i]))
d = np.log(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.log(data[i]))
d = np.log10(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.log10(data[i]))
d = np.log2(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.log2(data[i]))
d = np.sqrt(c)
for i, ei in enumerate(d):
self.assertEqual(ei, math.sqrt(data[i]))
# slices
data = [1., 2., 3.]
c = np.array(data + data)
d = np.cos(c[::2])
mm = data + data
for i, ei in enumerate(d):
self.assertEqual(ei, math.cos(mm[2 * i]))
# 2d array
data = [1., 2., 3.]
c = np.array([data, data])
d = np.cos(c)
mm = [data, data]
for i, ei in enumerate(d):
for j, eij in enumerate(ei):
self.assertEqual(eij, math.cos(mm[i][j]))
# 2d array slices
data = [1., 2., 3.]
c = np.array([data + data, data + data])
d = np.cos(c[:, ::2])
mm = [data + data, data + data]
for i, ei in enumerate(d):
for j, eij in enumerate(ei):
self.assertEqual(eij, math.cos(mm[i][2 * j]))
def gauss_cdf(z):
return 0.5 * (1 + np.erf(z / np.sqrt(2)))
def Phi(x):
""" cumulative gaussian fct """
return (np.erf(x * 0.7071067811865475) * 0.5 + 0.5)
def derived_error_function(N, P):
alpha = N / P
return 1 / 2 * (1 - np.erf((1 + alpha) / math.sqrt(2 * alpha)))
def gauss_cdf(z):
return 0.5 * (1 + np.erf(z / np.sqrt(2)))
def value(self, x): return np.erf(flattest(x)) def jacobian(self, x, into=None):
def V(x):
return - (k/16) * np.exp(-8 * (x ** 2)) - ((e2/(2 * (8 ** 0.5))) * np.sqrt(np.pi)) * (np.erf( (8 ** 0.5)* (x - xb)) - np.erf((8 ** 0.5) *( x + xb)))
