def calDCG(self, simM, label):
print "calculate DCG ..."
nCase = simM.shape[0]
DCG = 0
for i in range(nCase) :
labelI = label[i, 0]
nLabeI = np.where(label == labelI)[0].size
sortM = np.sort(simM[i, :])
index = [np.where(k == simM[i, :])[0].tolist()[0] for k in sortM]
G = []
[G.append(int(k == labelI)) for k in label[index,0]]
G = [G[j+1]*1.0/cmath.log(j+2, 2) for j, k in enumerate(G[1:])]
DCGI = reduce(lambda x,y :x+y, G, 1)
temp = [1.0/cmath.log(k,2) for k in range(2, nLabeI+1)]
DCGZ = reduce(lambda x,y : x+y, temp, 1)
DCG = DCG + DCGI*1.0/DCGZ
DCG = DCG*1.0/nCase
return DCG.real
def __init__(self, error_rate, elementNum):
# 计算所需要的bit数
self.bit_num = -1 * elementNum * cmath.log(error_rate) / (cmath.log(2.0) * cmath.log(2.0))
# 四字节对齐
self.bit_num = self.align_4byte(self.bit_num.real)
# 分配内存
self.bit_array = BitVector(size=self.bit_num)
# 计算hash函数个数
self.hash_num = cmath.log(2) * self.bit_num / elementNum
self.hash_num = self.hash_num.real
# 向上取整
self.hash_num = int(self.hash_num) + 1
# 产生hash函数种子
self.hash_seeds = self.generate_hashseeds(self.hash_num)
def __init__(self, error_rate, elementNum):
#计算所需要的 bit 数
self.bit_num = (-1) * elementNum * cmath.log(error_rate) / (cmath.log(2.0))
#四字节对齐
self.bit_num = self.align_4byte(self.bit_num.real)
#chxfan: CBF needs 4 times that BF need
self.bit_num = 4*self.bit_num
print ('bit_num = %d' % self.bit_num)
#分配内存
self.bit_array = BitVector(size=self.bit_num)
#计算 hash 函数个数, only effected by error_rate
self.hash_num = cmath.log(2) * self.bit_num / elementNum
self.hash_num = self.hash_num.real
#向上取整
self.hash_num = int(self.hash_num) + 1
print ('hash_num = %d' % self.hash_num)
#产生 hash 函数种子
self.hash_seeds = self.generate_hashseeds(self.hash_num)
def fieldValue(self, z): temp = (z-self.z1)*exp(-self.theta*1j) x = self.length/temp v1 = np.conjugate(-1.0j/2/pi*((1.0/x - 1)*log(1-x) + 1))*exp(self.theta*1j) v2 = np.conjugate(1.0j/2/pi*((1.0/x)*log(1-x) + 1))*exp(self.theta*1j) return (v1*self.strength1 + v2*self.strength2)
def matAValue(self, z): temp = (z-self.z1)*exp(-self.theta*1j) x = self.length/temp v1 = -1.0j/2/pi*((1.0/x - 1)*log(1-x) + 1) v1 = (v1.imag) v2 = 1.0j/2/pi*((1.0/x)*log(1-x) + 1) v2 = (v2.imag) return v1+v2
def etaSm(self, z, escTime, itr,c):
escTime = abs(escTime + 1 - (cmath.log( cmath.log(abs(z)) /
cmath.log(2) ) / cmath.log(2)))
return self.interpolate(escTime.real)
def test_logn(self):
import math, cmath
# log and log10 are tested in math (1:1 from rcomplex)
from numpy import log2, array, log1p
inf = float('inf')
ninf = -float('inf')
nan = float('nan')
cmpl = complex
log_2 = math.log(2)
a = [cmpl(-5., 0), cmpl(-5., -5.), cmpl(-5., 5.),
cmpl(0., -5.), cmpl(0., 0.), cmpl(0., 5.),
cmpl(-0., -5.), cmpl(-0., 0.), cmpl(-0., 5.),
cmpl(-0., -0.), cmpl(inf, 0.), cmpl(inf, 5.),
cmpl(inf, -0.), cmpl(ninf, 0.), cmpl(ninf, 5.),
cmpl(ninf, -0.), cmpl(ninf, inf), cmpl(inf, inf),
cmpl(ninf, ninf), cmpl(5., inf), cmpl(5., ninf),
cmpl(nan, 5.), cmpl(5., nan), cmpl(nan, nan),
]
for c,rel_err in (('complex128', 2e-15), ('complex64', 1e-7)):
b = log2(array(a,dtype=c))
for i in range(len(a)):
try:
_res = cmath.log(a[i])
res = cmpl(_res.real / log_2, _res.imag / log_2)
except OverflowError:
res = cmpl(inf, nan)
except ValueError:
res = cmpl(ninf, math.atan2(a[i].imag, a[i].real) / log_2)
msg = 'result of log2(%r(%r)) got %r expected %r\n ' % \
(c,a[i], b[i], res)
# cast untranslated boxed results to float,
# does no harm when translated
t1 = float(res.real)
t2 = float(b[i].real)
self.rAlmostEqual(t1, t2, rel_err=rel_err, msg=msg)
t1 = float(res.imag)
t2 = float(b[i].imag)
self.rAlmostEqual(t1, t2, rel_err=rel_err, msg=msg)
for c,rel_err in (('complex128', 2e-15), ('complex64', 1e-7)):
b = log1p(array(a,dtype=c))
for i in range(len(a)):
try:
#be careful, normal addition wipes out +-0j
res = cmath.log(cmpl(a[i].real+1, a[i].imag))
except OverflowError:
res = cmpl(inf, nan)
except ValueError:
res = cmpl(ninf, 0)
msg = 'result of log1p(%r(%r)) got %r expected %r\n ' % \
(c,a[i], b[i], res)
# cast untranslated boxed results to float,
# does no harm when translated
t1 = float(res.real)
t2 = float(b[i].real)
self.rAlmostEqual(t1, t2, rel_err=rel_err, msg=msg)
t1 = float(res.imag)
t2 = float(b[i].imag)
self.rAlmostEqual(t1, t2, rel_err=rel_err, msg=msg)
def test_faux_float(self):
class Foo:
def __float__(self):
return 1.0
class Bar(object):
def __float__(self):
return 1.0
self.assertEqual(cmath.log(Foo()), 0.0)
self.assertEqual(cmath.log(Bar()), 0.0)
def test_faux_complex(self):
class Foo:
def __complex__(self):
return 1.0j
class Bar(object):
def __complex__(self):
return 1.0j
self.assertEqual(cmath.log(Foo()), cmath.log(1.0j))
self.assertEqual(cmath.log(Bar()), cmath.log(1.0j))
def unitary_lerp(u1, u2, t):
"""
Continuously interpolates between 2x2 unitary matrices, with unitary intermediates.
:param u1: The initial unitary operation, used at t=0.
:param u2: The final unitary operation, used at t=1.
:param t: The interpolation factor, ranging from 0 to 1.
>>> np.sum(np.abs((unitary_lerp(np.mat([[1, 0], [0, 1]]), np.mat([[1, 0], [0, 1]]) * 1j, 0.5) \
- np.mat([[1, 0], [0, 1]]) * trig_tau.expi(1/8)))) < 0.000001
True
>>> (unitary_lerp(Rotation(x=0.5).as_pauli_operation(), Rotation(z=0.5).as_pauli_operation(), 0.5) \
== np.mat([[1,1],[1,-1]])/math.sqrt(2)).all()
True
>>> (unitary_lerp(Rotation(x=0.5).as_pauli_operation(), Rotation(x=0.5).as_pauli_operation(), 0.5) \
== Rotation(x=0.5).as_pauli_operation()).all()
True
>>> np.sum(np.abs(unitary_lerp(Rotation().as_pauli_operation(), Rotation(x=-0.25).as_pauli_operation(), 0.5) \
- Rotation(x=-0.125).as_pauli_operation())) < 0.000001
True
>>> np.sum(np.abs(unitary_lerp(Rotation().as_pauli_operation(), Rotation(x=0.75).as_pauli_operation(), 0.5) \
- Rotation(x=-0.125).as_pauli_operation())) < 0.000001
True
>>> np.sum(np.abs(unitary_lerp(Rotation().as_pauli_operation(), Rotation(x=0.25).as_pauli_operation(), 0.5) \
- Rotation(x=0.125).as_pauli_operation())) < 0.000001
True
>>> np.sum(np.abs(unitary_lerp(Rotation().as_pauli_operation(), Rotation(x=0.5).as_pauli_operation(), 0.5) \
- Rotation(x=0.25).as_pauli_operation())) < 0.000001
True
>>> np.sum(np.abs(unitary_lerp(Rotation(x=0.5).as_pauli_operation(), Rotation(x=0.75).as_pauli_operation(), 0.5) \
- Rotation(x=0.625).as_pauli_operation())) < 0.000001
True
"""
t1, x1, y1, z1, p1 = unitary_breakdown(u1)
t2, x2, y2, z2, p2 = unitary_breakdown(u2)
n1 = u1/p1
n2 = u2/p2
# Spherical interpolation of rotation part
dot = Quaternion(t1, x1, y1, z1).dot(Quaternion(t2, x2, y2, z2))
if dot < 0:
p2 *= -1
n2 *= -1
dot *= -1
theta = trig_tau.acos(max(min(dot, 1), -1))
c1 = trig_tau.sin_scale_ratio(theta, 1-t)
c2 = trig_tau.sin_scale_ratio(theta, t)
n3 = (n1*c1 + n2*c2)
# Angular interpolation of phase part
phase_angle_1 = cmath.log(p1).imag
phase_angle_2 = cmath.log(p2).imag
phase_drift = (phase_angle_2 - phase_angle_1 + math.pi) % trig_tau.tau - math.pi
phase_angle_3 = phase_angle_1 + phase_drift * t
p3 = cmath.exp(phase_angle_3 * 1j)
return n3 * p3
def _lanczos(z):
from cmath import pi, sin, log, exp
if z.real < 0:
return pi*z / (sin(pi*z) * _lanczos(-z))
else:
x = _lanczos_coef[0]
for i in range(1, 9):
x += _lanczos_coef[i]/(z+i)
logw = 0.91893853320467267+(z+0.5)*log(z+7.5)+log(x)-z-7.5
return exp(logw)
def __pow__(self, val):
ad_funcs = [self, to_auto_diff(val)] # list(map(to_auto_diff, (self, val)))
x = ad_funcs[0].x
y = ad_funcs[1].x
########################################
# Nominal value of the constructed ADF:
f = x**y
########################################
variables = self._get_variables(ad_funcs)
if not variables or isinstance(f, bool):
return f
########################################
# Calculation of the derivatives with respect to the arguments
# of f (ad_funcs):
if x.imag or y.imag:
if abs(x)>0 and ad_funcs[1].d(ad_funcs[1])!=0:
lc_wrt_args = [y*x**(y - 1), x**y*cmath.log(x)]
qc_wrt_args = [y*(y - 1)*x**(y - 2), x**y*(cmath.log(x))**2]
cp_wrt_args = x**(y - 1)*(y*cmath.log(x) + 1)/x
else:
lc_wrt_args = [y*x**(y - 1), 0.]
qc_wrt_args = [y*(y - 1)*x**(y - 2), 0.]
cp_wrt_args = 0.
else:
x = x.real
y = y.real
if x>0 and ad_funcs[1].d(ad_funcs[1])!=0:
lc_wrt_args = [y*x**(y - 1), x**y*math.log(x)]
qc_wrt_args = [y*(y - 1)*x**(y - 2), x**y*(math.log(x))**2]
cp_wrt_args = x**y*(y*math.log(x) + 1)/x
else:
lc_wrt_args = [y*x**(y - 1), 0.]
qc_wrt_args = [y*(y - 1)*x**(y - 2), 0.]
cp_wrt_args = 0.
########################################
# Calculation of the derivative of f with respect to all the
# variables (Variable) involved.
lc_wrt_vars, qc_wrt_vars, cp_wrt_vars = _apply_chain_rule(
ad_funcs, variables, lc_wrt_args,
qc_wrt_args, cp_wrt_args)
# The function now returns an ADF object:
return ADF(f, lc_wrt_vars, qc_wrt_vars, cp_wrt_vars)
def droste_effect(zoom_center_pixel_coords, zoom_factor, zoom_cutoff, source_image_filename, out_x_size = None, twist = False, zoom_loop_value = 0.0, save_filename = "sphere_transforms_test.png"):
"""produces a zooming Droste effect image from an equirectangular source image"""
# The source image should be a composite of the original image together with a zoomed version,
# fit inside a picture frame or similar in the original image
source_image = Image.open(source_image_filename)
s_im = source_image.load()
in_x_size, in_y_size = source_image.size
M_rot = rotate_pixel_coords_p_to_q(zoom_center_pixel_coords, (0,0), x_size = in_x_size)
M_rot_inv = matrix2_inv(M_rot)
if out_x_size == None:
out_x_size, out_y_size = source_image.size
else:
out_y_size = out_x_size/2
out_image = Image.new("RGB", (out_x_size, out_y_size))
o_im = out_image.load()
droste_factor = ( cmath.log(zoom_factor) + complex(0, 2*pi) ) / complex(0, 2*pi) # used if twisting
for i in range(out_x_size):
for j in range(out_y_size):
pt = (i,j)
pt = angles_from_pixel_coords(pt, x_size = out_x_size)
pt = sphere_from_angles(pt)
pt = CP1_from_sphere(pt)
pt = matrix_mult_vector(M_rot, pt)
# if ever you don't know how to do some operation in complex projective coordinates, it's almost certainly
# safe to just switch back to ordinary complex numbers by pt = pt[0]/pt[1]. The only danger is if pt[1] == 0,
# or is near enough to cause floating point errors. In this application, you are probably fine unless you
# make some very specific choices of where to zoom to etc.
pt = pt[0]/pt[1]
pt = cmath.log(pt)
if twist: # otherwise straight zoom
pt = droste_factor * pt
# zoom_loop_value is between 0 and 1, vary this from 0.0 to 1.0 to animate frames zooming into the droste image
pt = complex(pt.real + log(zoom_factor) * zoom_loop_value, pt.imag)
# zoom_cutoff alters the slice of the input image that we use, so that it covers mostly the original image, together with
# some of the zoomed image that was composited with the original. The slice needs to cover the seam between the two
# (i.e. the picture frame you are using, but should cover as little as possible of the zoomed version of the image.
pt = complex((pt.real + zoom_cutoff) % log(zoom_factor) - zoom_cutoff, pt.imag)
pt = cmath.exp(pt)
pt = [pt, 1] #back to projective coordinates
pt = matrix_mult_vector(M_rot_inv, pt)
pt = sphere_from_CP1(pt)
pt = angles_from_sphere(pt)
pt = pixel_coords_from_angles(pt, x_size = in_x_size)
o_im[i,j] = get_interpolated_pixel_colour(pt, s_im, in_x_size)
out_image.save(save_filename)
def fieldEffect(self,pos): """Velocity field due to this sheet on a pos""" z1=complex(self.x1[0],self.x1[1]) z2=complex(self.x2[0],self.x2[1]) zd=complex(pos[0],pos[1]) if (abs(z1-zd)<NODETOL) or (abs(z2-zd)<NODETOL): return(numpy.array([0.,0.])) z=(zd-z1)*(math.e**(-1j*self.theta)) pa1=self.gamma1*((((z/self.lemda)-1)*cmath.log((z-self.lemda)/z))+1) pa2=self.gamma2*((z/self.lemda*cmath.log((z-self.lemda)/z))+1) vel=1.j/2./math.pi*(pa1-pa2) vel=vel*(math.e**(-1j*self.theta)) return(numpy.array([vel.real,-vel.imag]))
def h(s, mq, mu):
"""Fermion loop function as defined e.g. in eq. (11) of hep-ph/0106067v2."""
if mq == 0.:
return 8/27. + (4j*pi)/9. + (8 * log(mu))/9. - (4 * log(s))/9.
if s == 0.:
return -4/9. * (1 + log(mq**2/mu**2))
z = 4 * mq**2/s
if z > 1:
A = atan(1/sqrt(z-1))
else:
A = log((1+sqrt(1-z))/sqrt(z)) - 1j*pi/2.
return (-4/9. * log(mq**2/mu**2) + 8/27. + 4/9. * z
-4/9. * (2 + z) * sqrt(abs(z - 1)) * A)
def arcinv(self, z, escTime, itr, c):
z = (cmath.log( cmath.log(z) /
cmath.log(2) ) / cmath.log(2))
r = abs(z.real)
g = abs(z.imag)
b = abs(z)
rgb = [i for i in [r,g,b]]
self.CSpace.setRGB(rgb)
self.CSpace.rgb2hsv()
self.CSpace.hsv[2] = sum(rgb)-self.CSpace.hsv[2]
return [int(255*(2-i)/2)%256 for i in self.CSpace.hsv2rgb()]
def plot_log(nor_distribution):
'''
Using normalized distribution to plot log/log graph
'''
degree_list = sorted(nor_distribution.keys())[1:]
value_list = []
for degree in degree_list:
value_list.append(nor_distribution[degree])
degree_x = [log(item) for item in degree_list]
value_y = [log(item) for item in value_list]
plt.plot(degree_x, value_y, 'ro')
plt.ylabel('log(distribution)')
plt.xlabel('log(degree)')
plt.show()
def __init__(self, error_rate, elementNum):
self.bit_num = -1 * elementNum * cmath.log(error_rate) / (cmath.log(2.0) * cmath.log(2.0))
self.bit_num = self.align_4byte(self.bit_num.real)
self.bit_array = BitVector(size=self.bit_num)
self.hash_num = cmath.log(2) * self.bit_num / elementNum
self.hash_num = self.hash_num.real
self.hash_num = int(self.hash_num) + 1
self.hash_seeds = self.generate_hashseeds(self.hash_num)
def __init__(self, amount = 1 << 26):
self.containerSize = (-1) * amount * cmath.log(0.001) / (cmath.log(2) * cmath.log(2)) #计算最佳空间大小
self.containerSize = int(self.containerSize.real) #取整
self.hashAmount = cmath.log(2) * (self.containerSize) / (amount)
self.hashAmount = int(self.hashAmount.real)
self.container = BitVector.BitVector(size = int(self.containerSize)) #分配内存
self.hashSeeds = find_prime(self.hashAmount)
self.hash = []
for i in range(int(self.hashAmount)): #生成哈希函数
self.hash.append(SimpleHash(self.containerSize, self.hashSeeds[i]))
return
def arg(complexe):
if isinstance(complexe, (int, complex, long, float)):
return _cmath.log(complexe).imag
elif isinstance(complexe, _sympy.Basic):
return _sympy.arg(complexe)
else:
return _numpy.imag(_numpy.log(complex))
def calcmsf(self, alpha, beta, transients=5, final=4000):
"""MSF ausrechnen.
"""
le = 0
self.params['alpha'] = alpha
self.params['beta'] = beta
system = neteqns(self.groups, self.H, self.T, self.eqns, params=self.params, coupling=self.coupling)
system.update(neteqns(self.groups, self.H, self.T, self.vareqns, params=self.params, coupling=self.coupling, var=True))
# print system # debug
for i in range(final):
sdde = dde23(eqns=system, params=self.params, debug=False)
sdde.set_sim_params(tfinal=self.params['tau'])
if i==0:
sdde.hist_from_arrays(self.__fillinitdict(self.transient, sdde.hist))
else:
norm = self.__calcnorm(linsol)
if i > transients:
le += log(norm)
sdde.hist_from_arrays(self.__renormalized(linsol, norm))
sdde.run()
sol = sdde.sol
linsol = sdde.sol_spl(self.resarray)
exponent = le/((i-transients)*self.params['tau'])
return exponent.real
def argand(base,save):
screen=pygame.display.set_mode([xsize,ysize])
if base.imag==0:
absv=int(abs(base.real)+0.5)
else:
absv= int(abs(base.real)**2+0.5) + int(abs(base.imag)**2+0.5)
if absv>96:
return
if absv==1:
raise Exception("No support for Base 1.")
limit=int(absv**(int(cmath.log(2**19,absv).real)))
checklist=[]
screen.fill((0,0,0))
maxlen=len(decToBase(limit,absv))
for n in range(0,limit):
oddBase=decToBase(n,absv)
new=baseToDec(oddBase,base)
#if new in checklist:
# raise Exception('Too many digits! Repeated at %s'%new)
#checklist+=[new]
#print("%s \t = %s"%(oddBase, new))
l=len(decToBase(n,absv))
x,y=int(new.real), int(new.imag)
cRatio=int((l+0.0)/maxlen*200)+55
screen.set_at((1*x+xsize/2,-1*y+ysize/2), (0,cRatio,cRatio))
new = "%s \t %s"%(new.real,new.imag)
pygame.display.flip()
if save:
pygame.image.save(screen, "<%s + %si>[0,%s^%s].jpeg"%(base.real, base.imag, absv,cmath.log(limit,absv).real))
def clngamma(z):
"""The logarithm of the gamma function for a complex argument
"""
z = complex(z)
# common case optimization
n = floor(z.real)
if n == z:
if n <= 0:
return complex(1.0 / 0.0) # Infinity
elif n <= len(__factorial):
return complex(__factorial[int(n) - 1])
zz = z
if z.real < 0.5:
zz = 1 - z
g = __lanczos_gamma
c = __lanczos_coefficients
zz = zz - 1.
zh = zz + 0.5
zgh = zh + g
zp = zgh ** (zh * 0.5) # trick for avoiding FP overflow above z=141
ss = 0.0
for k in range(len(c) - 1, 0, -1):
ss += c[k] / (zz + k)
f = (sqrt_2pi * (c[0] + ss)) * ((zp * cm.exp(-zgh)) * zp)
if z.real < 0.5:
f = pi / (cm.sin(pi * z) * f)
return cm.log(f)
def argand(base,save):
screen=pygame.display.set_mode([xsize,ysize])
if base.imag==0:
absv=int(abs(base.real))
else:
absv=int( base.real**2 + base.imag**2 )
if absv>96:
return
limit=int(abs(absv**(int(cmath.log(2**16,absv).real))))
checklist=[]
screen.fill((0,0,0))
maxlen=len(decToBase(limit,absv))
for n in range(limit):
oddBase=decToBase(n,absv)
new=baseToDec(oddBase,base)
#print("%s: %s"%(oddBase,new))
#if new in checklist:
# raise Exception("Two many digits")
#checklist+=[new]
l=len(decToBase(n,absv))
x,y=int(new.real), int(new.imag)
#Represent complex point as 2x2 box on argand diagram
pygame.draw.rect(screen,[0,int((l+0.0)/maxlen*200)+55,int((l+0.0)/maxlen*200)+55],[2*x+xsize/2,-2*y+ysize/2,2,2],0)
new = "%s \t %s"%(new.real,new.imag)
pygame.display.flip()
if save:
pygame.image.save(screen, "<%s + %si>[0,%s^%s].jpeg"%(base.real, base.imag, absv,cmath.log(limit,absv).real))
def cdigamma(z) :
"""Digamma function with complex arguments"""
z = complex(z)
g = __lanczos_gamma
c = __lanczos_coefficients
zz = z
if z.real < 0.5 :
zz = 1 -z
n=0.
d=0.
for k in range(len(c)-1,0,-1):
dz =1./(zz+(k+1)-2);
dd =c[k] * dz
d = d + dd
n = n - dd * dz
d = d + c[0]
gg = zz + g - 0.5
f = cm.log(gg) + (n/d - g/gg)
if z.real<0.5 :
f -= pi / cm.tan( pi * z)
return f
def log(a, b=math.e):
if not is_num(a) or not is_num(b):
raise BadTypeCombinationError(".l", a, b)
if a < 0:
return cmath.log(a, b)
return math.log(a, b)
def Hestf(phi, kappa, theta, sigma, rho, v0, r, T, s0, Type, q):
r = r-q
if (Type == 1):
u = 0.5
b = kappa - rho*sigma
else:
u = -0.5
b = kappa
a = kappa*theta
x = cm.log(s0)
d = cm.sqrt((rho*sigma*phi*1j-b)**2-sigma**2*(2*u*phi*1j-phi**2))
g = (b-rho*sigma*phi*1j + d)/(b-rho*sigma*phi*1j - d)
C = r*phi*1j*T + a/sigma**2*((b- rho*sigma*phi*1j + d)*T - 2*cm.log((1-g*cm.exp(d*T))/(1-g)))
D = (b-rho*sigma*phi*1j + d)/sigma**2*((1-cm.exp(d*T))/(1-g*cm.exp(d*T)))
return cm.exp(C + D*v0 + 1j*phi*x)
def log1p(x):
if x == -1:
return -inf
elif isinstance(x, complex) or x < -1:
return cmath.log(1 + x)
else:
return math.log1p(x)
def KL(a, b): if len(a) != len(b): return None result = 0 for x, y in zip(a, b): result += x * cmath.log(x/y) return result
def average(readings):
base = e ** (1j * tau / 360)
total = 0
for r in readings:
total += r[1] * base ** r[0]
result = total / len(readings)
return (log(result, base).real, abs(result))
def root(n, x):
if n == 0:
return math.nan
if x == 0 and n < 0:
return math.nan
if x < 0 and n % 2 == 0:
return math.nan
if x < 0:
result = cmath.exp((1/n) * cmath.log(x) )
r = math.sqrt(round(math.pow(result.real, 2.0) + math.pow(result.imag, 2.0)))
return -round(r)
else:
return math.pow(x, 1/n)
def do_func(self, funcstr, x):
return eval(
funcstr, {
"x": x,
"e": cmath.e,
"pi": cmath.pi,
"i": 1j,
"exp": cmath.exp,
"sin": cmath.sin,
"cos": cmath.cos,
"tan": cmath.tan,
"sinh": cmath.sinh,
"cosh": cmath.cosh,
"tanh": cmath.tanh,
"sec": lambda x: 1 / cmath.cos(x),
"csc": lambda x: 1 / cmath.sin(x),
"cot": lambda x: cmath.cos(x) / cmath.sin(x),
"sech": lambda x: 1 / cmath.cosh(x),
"csch": lambda x: 1 / cmath.sinh(x),
"coth": lambda x: cmath.cosh(x) / cmath.sinh(x),
"arcsin": cmath.asin,
"arccos": cmath.acos,
"arctan": cmath.atan,
"arsinh": cmath.asinh,
"arcosh": cmath.acosh,
"artanh": cmath.atanh,
"arcsec": lambda x: cmath.acos(1 / x),
"arccsc": lambda x: cmath.asin(1 / x),
"arccot": lambda x: cmath.atan(1 / x),
"arsech": lambda x: cmath.acosh(1 / x),
"arcsch": lambda x: cmath.asinh(1 / x),
"arcoth": lambda x: cmath.atanh(1 / x),
"abs": abs,
"sgn": sign,
"arg": cmath.phase,
"cis": lambda x: cmath.cos(x) + 1j * cmath.sin(x),
"pow": pow,
"sqrt": cmath.sqrt,
"nrt": lambda x, n: x**(1 / n),
"log": cmath.log,
"ln": lambda x: cmath.log(x),
"floor": math.floor,
"ceil": math.ceil,
"trunc": math.trunc,
"round": round,
"gamma": math_gamma,
"weierstrauss": math_weierstrauss,
"choose": math_choose,
"max": max,
"min": min
}, {})
def main():
start_clock = time.perf_counter()
delta = 0.1
Z2 = 0 # Z2数
for kx in np.arange(-pi, 0, delta):
print(kx)
for ky in np.arange(-pi, pi, delta):
H = hamiltonian(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 价带波函数1
vector2 = eigenvector[:, np.argsort(np.real(eigenvalue))[1]] # 价带波函数2
H_delta_kx = hamiltonian(kx+delta, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx的波函数1
vector_delta_kx2 = eigenvector[:, np.argsort(np.real(eigenvalue))[1]] # 略偏离kx的波函数2
H_delta_ky = hamiltonian(kx, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离ky的波函数1
vector_delta_ky2 = eigenvector[:, np.argsort(np.real(eigenvalue))[1]] # 略偏离ky的波函数2
H_delta_kx_ky = hamiltonian(kx+delta, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx和ky的波函数1
vector_delta_kx_ky2 = eigenvector[:, np.argsort(np.real(eigenvalue))[1]] # 略偏离kx和ky的波函数2
Ux = dot_and_det(vector, vector_delta_kx, vector2, vector_delta_kx2)
Uy = dot_and_det(vector, vector_delta_ky, vector2, vector_delta_ky2)
Ux_y = dot_and_det(vector_delta_ky, vector_delta_kx_ky, vector_delta_ky2, vector_delta_kx_ky2)
Uy_x = dot_and_det(vector_delta_kx, vector_delta_kx_ky, vector_delta_kx2, vector_delta_kx_ky2)
F = np.imag(cmath.log(Ux*Uy_x*np.conj(Ux_y)*np.conj(Uy)))
A = np.imag(cmath.log(Ux))+np.imag(cmath.log(Uy_x))+np.imag(cmath.log(np.conj(Ux_y)))+np.imag(cmath.log(np.conj(Uy)))
Z2 = Z2 + (A-F)/(2*pi)
print('Z2 = ', Z2) # Z2数
end_clock = time.perf_counter()
print('CPU执行时间(min)=', (end_clock-start_clock)/60)
def average(readings):
base = e ** (1j * tau / 360)
total = 0
for r in readings:
v = r[1] * base ** r[0]
total += v
arrow(0, 0, v.real, v.imag, head_width=0.05, head_length=0.05, fc='r', ec='r')
result = total / len(readings)
arrow(0, 0, result.real, result.imag, head_width=0.05, head_length=0.05, fc='b', ec='b')
xlim((-1.5, 1.5))
ylim((-1.5, 1.5))
ylabel('Imaginary')
xlabel('Real')
return (cmath.log(result, base).real, abs(result))
def log(self, base=math.e):
# TODO: is this correct?
if not self.front:
raise ValueError('cannot take log of 0')
if self.leading < 0:
raise ValueError('cannot take log of infinite')
elif self.leading > 0:
raise ValueError('cannot take log of infinitesimal')
if isinstance(base, LeviCivitaBase):
return self.log() / base.log()
if base != math.e:
return self.log() / cmath.log(base)
series = (i * -1**(i - 1) for i in itertools.count())
return self._MATH.log(self.front) + self.eps_part().expand(series)
def _logarithmic_func(self) -> None:
number: Union[float, complex] = self._number_on_screen
self._operation_str_var.set("ln(x)")
if isinstance(number, complex):
self._number_screen.set(cmath.log(number))
else:
if number > 0:
self._number_screen.set(math.log(number))
else:
self._number_screen.set("USE ONLY POSITIVE NUMBERS")
self._reset_screen = True
def test10_math_explog():
for i in range(-5, 5):
for j in range(-5, 5):
if i != 0 or j != 0:
a = ek.log(C(i, j))
b = C(cmath.log(complex(i, j)))
assert ek.allclose(a, b)
a = ek.log2(C(i, j))
b = C(cmath.log(complex(i, j)) / cmath.log(2))
assert ek.allclose(a, b)
a = ek.exp(C(i, j))
b = C(cmath.exp(complex(i, j)))
assert ek.allclose(a, b)
a = ek.exp2(C(i, j))
b = C(cmath.exp(complex(i, j) * cmath.log(2)))
assert ek.allclose(a, b)
a = ek.pow(C(2, 3), C(i, j))
b = C(complex(2, 3)**complex(i, j))
assert ek.allclose(a, b)
def __init__(self, error_rate=0.001, elementNum=10000000, lock=None):
#计算所需要的bit数
self.bit_num = -1 * elementNum * cmath.log(error_rate) / (cmath.log(2.0) * cmath.log(2.0))
#四字节对齐
self.bit_num = self.align_4byte(self.bit_num.real)
#分配内存
self.bit_array = BitVector(size=self.bit_num)
#计算hash函数个数
self.hash_num = cmath.log(2) * self.bit_num / elementNum
self.hash_num = self.hash_num.real
#向上取整
self.hash_num = int(self.hash_num) + 1
#产生hash函数种子
self.hash_seeds = self.generate_hashseeds(self.hash_num)
#锁
self.lock = lock
def BM25(index, nfiles, docName, QueryWords, AVG_L, LD):
score = 0
docName = str(docName)
k1 = 1.2
b = 0.75
for word in QueryWords:
if word not in index or docName not in index[word]:
continue
tf = len(index[word][docName])
df = len(index[word])
w = cmath.log(2, (nfiles - df + 0.5) / (df + 0.5)).real
s = (k1 * tf * w) / (tf + k1 * (1 - b + b * LD[docName] / AVG_L))
score += s
return score
def calcShannonent(dataSet):
num_entyies = len(dataSet)
labelCounts = {}
for featvec in dataSet:
currentLabel = featvec[-1]
if currentLabel not in labelCounts.keys():
labelCounts[currentLabel] = 0
labelCounts[currentLabel] += 1
print(labelCounts)
shannonEnt = 0.0
for key in labelCounts:
prob = float(labelCounts[key]) / num_entyies
shannonEnt -= prob * log(prob, 2) # log base 2
return shannonEnt
def testRLGCExtract(self):
return
sp = si.sp.SParameterFile('cableForRLGC.s2p')
Z0 = 50
Tdest = 2.24e-9
fList = sp.m_f[1:]
S11 = sp.Response(1, 1)[1:]
S12 = sp.Response(1, 2)[1:]
ra = [(s11 * s11 + 1 - s12 * s12) / (2 * s11)
for (s11, s12) in zip(S11, S12)]
rb = [
cmath.sqrt(s11 * s11 * s11 * s11 - 2 * s11 * s11 - 2 * s11 * s11 *
s12 * s12 + 1 - 2 * s12 * s12 + s12 * s12 * s12 * s12) /
(2 * s11) for (s11, s12) in zip(S11, S12)
]
rho = [ra - rb for (ra, rb) in zip(ra, rb)]
la = [(r * r - 1) / (2 * s12 * r * r) for (r, s12) in zip(rho, S12)]
lb = [
cmath.sqrt(1 - 2 * r * r + r * r * r * r + 4 * s12 * s12 * r * r) /
(2 * s12 * r * r) for (r, s12) in zip(rho, S12)
]
Ls = [la + lb for (la, lb) in zip(la, lb)]
Zc = [-Z0 * (r + 1) / (r - 1) for r in rho]
gamma = [
-(cmath.log(l * cmath.exp(1j * 2 * math.pi * f * Tdest)) -
1j * 2 * math.pi * f * Tdest) for (l, f) in zip(Ls, fList)
]
Y = [g / zc for (g, zc) in zip(gamma, Zc)]
Z = [g * zc for (g, zc) in zip(gamma, Zc)]
Rr = [abs(z.real) for z in Z]
L = [z.imag / (2. * math.pi * f) for (z, f) in zip(Z, fList)]
Gr = [abs(y.real) for y in Y]
C = [y.imag / (2. * math.pi * f) for (y, f) in zip(Y, fList)]
print(Zc)
import matplotlib.pyplot as plt
plt.clf()
plt.title('s-parameter compare')
plt.xlabel('frequency (GHz)')
plt.ylabel('amplitude')
plt.semilogy([f / 1e9 for f in fList], C, label='C')
plt.semilogy([f / 1e9 for f in fList], L, label='L')
plt.semilogy([f / 1e9 for f in fList], Rr, label='R')
plt.semilogy([f / 1e9 for f in fList], Gr, label='G')
plt.legend(loc='upper right')
plt.grid(True)
plt.show()
(Rx, Rsex, Lx) = (matrix([[1., math.sqrt(f), 1j * 2 * math.pi * f]
for f in fList]).getI() *
matrix([[z] for z in Z])).tolist()
print((Rx[0].real, Rsex[0].real, Lx[0].real))
def Heston_CharacteristicFunctionLogPrice(t, omega, S0, v0, r, kappa, theta,
ksi, rho):
alpha = complex(-omega**2 / 2, -omega / 2)
beta = complex(kappa, -rho * ksi * omega)
gamma = ksi**2 / 2
h = (beta**2 - 4 * alpha * gamma)**0.5
r_m = (beta - h) / ksi**2
r_p = (beta + h) / ksi**2
g = r_m / r_p
C = kappa * (r_m * t - 2 * cpx.log(
(1 - g * cpx.exp(-h * t)) / (1 - g)) / ksi**2)
D = r_m * ((1 - cpx.exp(-h * t)) / (1 - g * cpx.exp(-h * t)))
return cpx.exp(
complex(C * theta + D * v0, omega * np.log(S0 * np.exp(r * t))))
def test_cmath_matches_math(self):
# check that corresponding cmath and math functions are equal
# for floats in the appropriate range
# test_values in (0, 1)
test_values = [0.01, 0.1, 0.2, 0.5, 0.9, 0.99]
# test_values for functions defined on [-1., 1.]
unit_interval = test_values + [-x for x in test_values] + \
[0., 1., -1.]
# test_values for log, log10, sqrt
positive = test_values + [1.] + [1. / x for x in test_values]
nonnegative = [0.] + positive
# test_values for functions defined on the whole real line
real_line = [0.] + positive + [-x for x in positive]
test_functions = {
'acos': unit_interval,
'asin': unit_interval,
'atan': real_line,
'cos': real_line,
'cosh': real_line,
'exp': real_line,
'log': positive,
'log10': positive,
'sin': real_line,
'sinh': real_line,
'sqrt': nonnegative,
'tan': real_line,
'tanh': real_line
}
for fn, values in test_functions.items():
float_fn = getattr(math, fn)
complex_fn = getattr(cmath, fn)
for v in values:
z = complex_fn(v)
if not due_to_ironpython_bug(
"http://ironpython.codeplex.com/workitem/28352"):
self.rAssertAlmostEqual(float_fn(v), z.real)
self.assertEqual(0., z.imag)
# test two-argument version of log with various bases
for base in [0.5, 2., 10.]:
for v in positive:
z = cmath.log(v, base)
self.rAssertAlmostEqual(math.log(v, base), z.real)
self.assertEqual(0., z.imag)
def addNewEntry(position, isZero):
entry = RootListEntry(entryFrame)
entry.label.bind("<Button-1>", entrySelect)
entry.pairButton.bind("<Button-1>", entryPair)
entry.removeButton.bind("<Button-1>", entryRemove)
# organize existing list of entries
for entryIndex in range(0, len(entryFrame.winfo_children())):
entryFrame.winfo_children()[entryIndex].grid(row = entryIndex)
if (isZero):
entryRoot = Zero(position)
zPlaneRoot = DrawableZero(zPlaneComplex, entryRoot.getPosition())
sPlaneRoot = DrawableZero(sPlaneComplex, cmath.log(entryRoot.getPosition()))
else:
entryRoot = Pole(position)
zPlaneRoot = DrawablePole(zPlaneComplex, entryRoot.getPosition())
sPlaneRoot = DrawablePole(sPlaneComplex, cmath.log(entryRoot.getPosition()))
entry.label.config(text=repr(entryRoot.getPosition()))
key = id(entry)
entryDictionary[key] = EntryElements(zPlaneRoot, sPlaneRoot, entryRoot)
def poly_num_roots(f, Df, circle, N):
'''poly_num_roots(f, f', A->B->.., N) == residue(f'/f, A->B->.., N)
bug: if circle not convex or outside roots too near
we can cut the z-plane into strikes by R+y*i for some real R.
note that each root contribute to +/-pi over such line.
sadly, log is not continue over all plane, we have to do numerial integrate.
how to trace the change of angle????????????????
'''
Df_f = lambda z: Df(z) / f(z)
circle = list(circle)
i = 1
while i < len(circle):
if circle[i - 1] != circle[i]: break
i += 1
circle = circle[i - 1:]
res = round(residue(Df_f, circle, N))
path = circle + circle[:1]
# watch the change of angle of f(z), that is imag of log(f(z))
ss = []
for a, b in zip(path, path[1:]):
L = b - a
dL = L / N
s = (f(a + L * t / N) for t in range(1, N + 1))
angles = (cmath.log(v).imag for v in s)
ss.extend(angles) # angle counts
ss.append(ss[0])
if ss[0] == 0 or ss[0] == 2 * math.pi:
if len(circle) > 2 or len(circle) == 2 and N > 1:
ss.append(ss[1])
#print('ss.append')
#print('angles[0]=={}'.format(ss[0]))
print('angles[:10]=={}'.format(ss[:10]))
#print('angles[-10:]=={}'.format(ss[-10:]))
num_roots1 = sum(a > b for a, b in zip(ss, ss[1:]))
num_roots2 = len(ss) - num_roots1
num_roots = min(num_roots1, num_roots2)
#num_roots = num_roots1
if num_roots != res:
print('num_roots != res', num_roots, res)
assert abs(num_roots - res) < 2
return num_roots
def F_87(Lmu, sh):
"""Function $F_8^{(7)}$ giving the contribution of $O_7$ to the matrix element
of $O_8$, as given in eq. (40) of hep-ph/0312063.
- `sh` is $\hat s=q^2/m_b^2$,
"""
flavio.citations.register("Asatrian:2003vq")
if sh == 0.:
return (-4 * (33 + 24 * Lmu + 6j * pi - 2 * pi**2)) / 27.
return (-32 / 9. * Lmu + 8 / 27. * pi**2 - 44 / 9. - 8 / 9. * 1j * pi +
(4 / 3. * pi**2 - 40 / 3.) * sh +
(32 / 9. * pi**2 - 316 / 9.) * sh**2 +
(200 / 27. * pi**2 - 658 / 9.) * sh**3 - 8 / 9. * log(sh) *
(sh + sh**2 + sh**3))
def updateEntryPosition(entry, position):
position = round(position.real, 5) + round(position.imag, 5) * 1j
key = id(entry)
entryDictionary[key].complexRoot.setPosition(position)
position = entryDictionary[key].complexRoot.getPosition()
zPlanePoint = position
sPlanePoint = cmath.log(position)
entryDictionary[key].zPlaneDrawable.setPosition(zPlanePoint)
entryDictionary[key].sPlaneDrawable.setPosition(sPlanePoint)
entry.label.config(text=repr(position))
def get_velocity(self, location):
res = 0
for i in range(len(self.panel_points) - 1):
p1 = self.panel_points[i]
p2 = self.panel_points[i + 1]
relative_vector = p2.location - p1.location
t = (relative_vector) / abs(relative_vector)
tc = t.conjugate()
z = (location - p1.location) * tc
uiv = (1j * p1.gamma *
cmath.log(1 - (abs(relative_vector) / z))) / (2 * np.pi)
uiv = uiv.conjugate()
uiv = uiv * t
res += uiv
return res
def calcShannonEnt(dataSet):
numEntries = len(dataSet) #数据集的个数
labelCounts = {} #存储每一类标签和出现的次数，为后面计算香农熵
for fentVec in dataSet:
currentLabel = fentVec[-1] #最后一列作为标签
#labelCounts[currentLabel] = labelCounts.get(currentLabel,0) + 1 #这行代码可以由以下三行代替
if currentLabel not in labelCounts.keys():
labelCounts[currentLabel] = 0
labelCounts[currentLabel] += 1
shannonEnt = 0.0 #初始化香农熵
for key in labelCounts:
prob = float(labelCounts[key] / numEntries) #每一类标签出现的概率
shannonEnt -= prob * (log(prob,
2)).real #计算信息熵 ，log函数返回的是实数+虚数的格式，2表示以2为底
return shannonEnt #熵越高混合的数据就越多
def complex_potential(self, z: complex) -> complex:
"""
Well's complex potential at location <z>.
Return the well's contribution to the complex potential,
Omega(z) [L^3/T], evaluated at location <z>.
Arguments:
z (complex): 'little z' world coordinate location [L].
Returns:
complex: complex potential at location <z> [L^3/T].
Notes:
- If the location <z> is inside the radius of the well, the
complex potential at the radius of the well is returned.
"""
zz = z - self.z
if abs(zz) >= self.r:
Omega = self.Q / (2 * cmath.pi) * cmath.log(zz)
else:
Omega = self.Q / (2 * cmath.pi) * cmath.log(self.r)
return Omega
def LogLikelihood(para, x_array):
"""
para: an array of Meixner parameters
x_array: your observation array
"""
n = len(x_array) #number of your observation
meixner = [Meixner(para, x_array[i]) for i in range(n)]
#log likelihood function
#LL=cmath.log(np.multiply.reduce(meixner)) #this may cause overflow
single_LL = [cmath.log(i) for i in meixner if i.real > 0]
LL = sum(single_LL)
#LL=np.multiply.reduce(meixner)
return -LL.real
def makeLocalExpansion(self, box):
"""Lemma 4.7 (Conversion of a multipole expansion into a local expansion)"""
zo = (box.x-self.x) + (box.y-self.y)*1j
# Equation 4.18 - calculating b0
self.bl[0] += box.ak[0]*log(-1.0*zo)
for k in range(1,self.p):
self.bl[0] += box.ak[k]*((-1.0)**k)/(zo**k)
# Equation 4.19 - calculation bl
for l in range(1,self.p):
self.bl[l] += -1.0*box.ak[0]/(l*(zo**l))
tmp = 0.0
for k in range(1,self.p):
tmp += ((box.ak[k])/(zo**k))*((-1.0)**k)*bincoeff(l+k-1,k-1)
self.bl[l] += (1/(zo**l))*tmp
return None
def calc(self):
'''
Within this calculation the bypass ratio of the engine is determined from the engine diameter and the
takeoff thurst of the engine. It is questionable whether this is a suitable approach as it probably breaks with the underlying statistic.
:Source: Aircraft Design: A Conceptual Approach, D. P. Raymer, AIAA Education Series, 1992, Second Edition, p. 198, Eq. 10.6
'''
Tto = self.parent.thrustTO.getValue()
d = self.parent.dEngine.getValue()
return self.setValueCalc(0.25 * (log(d / Tto) + 5.351))
###################################################################################################
#EOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFEOFE#
###################################################################################################
def ewlc(inp, p0, p1, p2, p3): # p0=Lk p1=Ks p2=Ns p3=Lc
output_E = []
for f in inp:
extension1 = p2 * (Lplanar / (np.exp(-deltaG / kT) + 1) + Lhelical /
(np.exp(deltaG / kT) + 1)) * (
coth(p0 * f / kT) - kT / (p0 * f)) + p2 * f / p1
Q = 1 + f / K
P = f / K + Lp * f / kT + 1 / 4
l = -2 * Q - P
u = (-1 / 9) * (Lp * f / kT - 3 / 4)**2
v = (-1 / 27) * (Lp * f / kT - 3 / 4)**3 + (1 / 8) + 0.0000000000001
if (v**2 + u**3) < 0:
theta = np.arccos(np.sqrt(-1 * v**2 / u**3))
if v < 0:
result = extension1 + (
2 * np.sqrt(-u) * np.cos(theta / 3 + 2 * np.pi / 3) -
l / 3) * p3
else:
result = extension1 + (
-2 * np.sqrt(-u) * np.cos(theta / 3 - 6 * np.pi / 3) -
l / 3) * p3
else:
A = cmath.exp(
(1 / 3) * cmath.log(-v + np.sqrt(1 / 64 - 1 / 4 * complex(
np.power((Lp * f / kT - 3 / 4) / 3, 3), 0))))
B = cmath.exp(
(1 / 3) * cmath.log(-v - np.sqrt(1 / 64 - 1 / 4 * complex(
np.power((Lp * f / kT - 3 / 4) / 3, 3), 0))))
E = -np.abs((A + B)) - l / 3
result = extension1 + E * p3
output_E.append(result)
return output_E
def forward(self, x):
with torch.no_grad():
def wf(psi, x):
(a, b) = psi.forward(x)
return cmath.exp(complex(a, b))
avg_psi = np.mean(np.array([wf(psi, x) for psi in self._machines]))
avg_log_psi = cmath.log(avg_psi)
x = torch.tensor(
[avg_log_psi.real, avg_log_psi.imag],
dtype=torch.float32,
requires_grad=False,
)
return x
def main():
start_clock = time.perf_counter()
delta = 0.005
chern_number = 0 # 陈数初始化
# 几个坐标中常出现的项
a = 1/sqrt(3)
aa1 = 4*sqrt(3)*pi/9/a
aa2 = 2*sqrt(3)*pi/9/a
bb1 = 2*pi/3/a
hamiltonian0 = functools.partial(hamiltonian, M=2/3, t1=1, t2=1/3, phi=pi/4, a=a) # 使用偏函数，固定一些参数
for kx in np.arange(-aa1, aa1, delta):
print(kx)
for ky in np.arange(-bb1, bb1, delta):
if (-aa2<=kx<=aa2) or (kx>aa2 and -(aa1-kx)*tan(pi/3)<=ky<=(aa1-kx)*tan(pi/3)) or (kx<-aa2 and -(kx-(-aa1))*tan(pi/3)<=ky<=(kx-(-aa1))*tan(pi/3)): # 限制在六角格子布里渊区内
H = hamiltonian0(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 价带波函数
H_delta_kx = hamiltonian0(kx+delta, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx的波函数
H_delta_ky = hamiltonian0(kx, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离ky的波函数
H_delta_kx_ky = hamiltonian0(kx+delta, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx和ky的波函数
Ux = np.dot(np.conj(vector), vector_delta_kx)/abs(np.dot(np.conj(vector), vector_delta_kx))
Uy = np.dot(np.conj(vector), vector_delta_ky)/abs(np.dot(np.conj(vector), vector_delta_ky))
Ux_y = np.dot(np.conj(vector_delta_ky), vector_delta_kx_ky)/abs(np.dot(np.conj(vector_delta_ky), vector_delta_kx_ky))
Uy_x = np.dot(np.conj(vector_delta_kx), vector_delta_kx_ky)/abs(np.dot(np.conj(vector_delta_kx), vector_delta_kx_ky))
F = cmath.log(Ux*Uy_x*(1/Ux_y)*(1/Uy))
# 陈数(chern number)
chern_number = chern_number + F
chern_number = chern_number/(2*pi*1j)
print('Chern number = ', chern_number)
end_clock = time.perf_counter()
print('CPU执行时间(min)=', (end_clock-start_clock)/60)
def laguer(coef, m, x):
# Returns a single coefficient
MAXIT = 80
MT = 10
EPSS = 1.0e-7
frac = [0.0, 0.5, 0.25, 0.75, 0.13, 0.38, 0.62, 0.88, 1.0]
for iter in range(1, MAXIT+1):
b = coef[m]
err = abs(b)
d = 0+0j
f = 0+0j
abx = abs(x)
for j in range(m-1, -1, -1):
f = d + x*f
d = b + x*d
b = coef[j] + x*b
err = abs(b) + abx*err
err *= EPSS
if abs(b) <= err:
return x
g = d / b
g2 = g * g
h = g2 - 2.0*(f/b)
sq = cmath.sqrt(float(m-1)*(m*h - g2))
gp = g + sq
gm = g - sq
abp = abs(gp)
abm = abs(gm)
if (abp < abm):
gp = gm
if max(abp, abm) > 0.0:
dx = (m+0.0j) / gp
else:
dx = cmath.exp(cmath.log(1+abx)) * complex(math.cos(float(iter)),
math.sin(float(iter)))
x1 = x - dx
if x.real == x1.real and x.imag == x1.imag:
return x
if iter % MT:
x = x1
else:
x = x - frac[iter/MT] * dx
print "too many iterations in laguerre"
return x
def cbrt(x):
"""cube root of x; does not return negative numbers
Examples:
------------
>>> cbrt(-8)
-2.0
>>> cbrt(8)
2.0
"""
if isinstance(x, Real):
return copysign(pow(abs(x), 1/3), x.real)
elif isinstance(x, complex):
return exp((1/3)*log(x))
else:
raise TypeError('x must be a number')
def _get_complex_log_weight(self):
"""
Get the log weight even in cases where the determinant of the covariance matrix is negative. In such cases the
log_weight no longer corresponds to the integral and the log of the weight will have a imaginary component.
Computing the complex log-weight can however still be useful: it is used, for example, in the (experimental)
Gaussian mixture division function (_gm_division_m2).
:return: The potentially complex log weight
:rtype: complex
"""
self._update_canform()
cov = np.linalg.inv(self.prec)
mean = cov @ self.h_vec
ut_prec_u = (mean.T @ self.prec @ mean).astype(complex)
log_weight = self.g_val + 0.5 * ut_prec_u + 0.5 * cmath.log(np.linalg.det(2.0 * np.pi * cov))
return log_weight
def find_z(p, q):
#print(p,q)
P = p**3 / 27
Q = q**2 / 4
R = Q + P
if (abs(Q and R/Q or R)) < 1e-13:
R = 0
if R < 0:
# z^3 is complex, so z will be too (conjugate pair)
z = exp(log(-q/2 + sqrt(R)) / 3)
return z, z.conjugate()
if R == 0:
z = cuberoot(-q/2)
return z, z
rR = sqrt(R).real
return cuberoot(-q/2+rR), cuberoot(-q/2-rR)