def simulateHestonModel( T, N, R, mu, kappa, vBar, sigmaV, rho, x0, v0 ) :
deltaT = T / (float)(N - 1)
x = [af.constant(x0, R, dtype=af.Dtype.f32), af.constant(0, R, dtype=af.Dtype.f32)]
v = [af.constant(v0, R, dtype=af.Dtype.f32), af.constant(0, R, dtype=af.Dtype.f32)]
sqrtDeltaT = math.sqrt(deltaT)
sqrtOneMinusRhoSquare = math.sqrt(1-rho**2)
m = af.constant(0, 2, dtype=af.Dtype.f32)
m[0] = rho
m[1] = sqrtOneMinusRhoSquare
zeroArray = af.constant(0, R, 1, dtype=af.Dtype.f32)
for t in range(1, N) :
tPrevious = (t + 1) % 2
tCurrent = t % 2
dBt = af.randn(R, 2, dtype=af.Dtype.f32) * sqrtDeltaT
vLag = af.maxof(v[tPrevious], zeroArray)
sqrtVLag = af.sqrt(vLag)
x[tCurrent] = x[tPrevious] + (mu - 0.5 * vLag) * deltaT + sqrtVLag * dBt[:, 0]
v[tCurrent] = vLag + kappa * (vBar - vLag) * deltaT + sigmaV * (sqrtVLag * af.matmul(dBt, m))
return (x[tCurrent], af.maxof(v[tCurrent], zeroArray))
def black_scholes(S, X, R, V, T):
# S = Underlying stock price
# X = Strike Price
# R = Risk free rate of interest
# V = Volatility
# T = Time to maturity
d1 = af.log(S / X)
d1 = d1 + (R + (V * V) * 0.5) * T
d1 = d1 / (V * af.sqrt(T))
d2 = d1 - (V * af.sqrt(T))
cnd_d1 = cnd(d1)
cnd_d2 = cnd(d2)
C = S * cnd_d1 - (X * af.exp((-R) * T) * cnd_d2)
P = X * af.exp((-R) * T) * (1 - cnd_d2) - (S * (1 -cnd_d1))
return (C, P)
def band_energy(p1, p2):
if (p_space_grid == 'cartesian'):
p_x = p1
p_y = p2
elif (p_space_grid == 'polar2D'):
p_x = p1 * af.cos(p2)
p_y = p1 * af.sin(p2)
else:
raise NotImplementedError('Unsupported coordinate system in p_space')
p = af.sqrt(p_x**2. + p_y**2.)
E_upper = p * fermi_velocity
af.eval(E_upper)
return (E_upper)
def initialize_f(q1, q2, p1, p2, p3, params):
m = params.mass
k = params.boltzmann_constant
rho_b = params.rho_background
T_b = params.temperature_background
p1_bulk = params.p1_bulk_background
p2_bulk = params.p2_bulk_background
p3_bulk = params.p3_bulk_background
pert_real = params.pert_real
pert_imag = params.pert_imag
k_q1 = params.k_q1
k_q2 = params.k_q2
# Calculating the perturbed density:
rho = rho_b + (pert_real * af.cos(k_q1 * q1 + k_q2 * q2) -
pert_imag * af.sin(k_q1 * q1 + k_q2 * q2))
# Depending on the dimensionality in velocity space, the
# distribution function is assigned accordingly:
if (params.p_dim == 3):
f = rho * (m / (2 * np.pi * k * T_b))**(3 / 2) \
* af.exp(-m * (p1 - p1_bulk)**2 / (2 * k * T_b)) \
* af.exp(-m * (p2 - p2_bulk)**2 / (2 * k * T_b)) \
* af.exp(-m * (p3 - p3_bulk)**2 / (2 * k * T_b))
elif (params.p_dim == 2):
f = rho * (m / (2 * np.pi * k * T_b)) \
* af.exp(-m * (p1 - p1_bulk)**2 / (2 * k * T_b)) \
* af.exp(-m * (p2 - p2_bulk)**2 / (2 * k * T_b))
else:
f = rho * af.sqrt(m / (2 * np.pi * k * T_b)) \
* af.exp(-m * (p1 - p1_bulk)**2 / (2 * k * T_b))
af.eval(f)
return (f)
def band_velocity(p1, p2):
# Note :This function is only meant to be called once to initialize the vel vectors
p_x, p_y = get_p_x_and_p_y(p1, p2)
p = af.sqrt(p_x**2. + p_y**2.)
p_hat = [p_x / (p + 1e-20), p_y / (p + 1e-20)]
if (fermi_surface_shape == 'circle'):
v_f_hat = p_hat
elif (fermi_surface_shape == 'hexagon'):
# Need to get theta for normal_to_hexagon_unit_vec()
if (p_space_grid == 'cartesian'):
p_x_local = p1
p_y_local = p2
theta = af.atan(p_y_local / p_x_local)
elif (p_space_grid == 'polar2D'):
# In polar2D coordinates, p1 = radius and p2 = theta
r = p1
theta = p2
else:
raise NotImplementedError(
'Unsupported coordinate system in p_space')
v_f_hat = normal_to_hexagon_unit_vec(theta)
# Quadratic dispersion
m = effective_mass(p1, p2)
v_f = p / m
if (dispersion == 'linear' or zero_temperature):
v_f = fermi_velocity
upper_band_velocity = [v_f * v_f_hat[0], v_f * v_f_hat[1]]
return (upper_band_velocity)
def sqrt_det_g(q1, q2, q1_start_local_left=None, q2_start_local_bottom=None):
jac = jacobian_dx_dq(q1, q2, q1_start_local_left, q2_start_local_bottom)
dx_dq1 = jac[0][0]
dx_dq2 = jac[0][1]
dy_dq1 = jac[1][0]
dy_dq2 = jac[1][1]
g_11 = (dx_dq1)**2. + (dy_dq1)**2.
g_12 = dx_dq1 * dx_dq2 + dy_dq1 * dy_dq2
g_21 = g_12
g_22 = (dx_dq2)**2. + (dy_dq2)**2.
det_g = g_11 * g_22 - g_12 * g_21
return (af.sqrt(det_g))
def band_velocity(p1, p2):
if (p_space_grid == 'cartesian'):
p_x = p1
p_y = p2
elif (p_space_grid == 'polar2D'):
p_x = p1 * af.cos(p2)
p_y = p1 * af.sin(p2)
else :
raise NotImplementedError('Unsupported coordinate system in p_space')
p = af.sqrt(p_x**2. + p_y**2.)
p_hat = [p_x / (p + 1e-20), p_y / (p + 1e-20)]
v_f = fermi_velocity
upper_band_velocity = [ v_f * p_hat[0], v_f * p_hat[1]]
af.eval(upper_band_velocity[0], upper_band_velocity[1])
return(upper_band_velocity)
def sqrt_det_g(q1, q2):
jac = jacobian_dx_dq(q1, q2)
dx_dq1 = jac[0][0]
dx_dq2 = jac[0][1]
dy_dq1 = jac[1][0]
dy_dq2 = jac[1][1]
g_11 = (dx_dq1)**2. + (dy_dq1)**2.
g_12 = dx_dq1 * dx_dq2 + dy_dq1 * dy_dq2
g_21 = g_12
g_22 = (dx_dq2)**2. + (dy_dq2)**2.
det_g = g_11 * g_22 - g_12 * g_21
return (af.sqrt(det_g))
def initialize_f(q1, q2, p1, p2, p3, params):
m = params.mass_particle
k = params.boltzmann_constant
rho_b = params.rho_background
T_b = params.temperature_background
# Calculating the perturbed density:
rho = rho_b + 0.01 * af.exp(-(q1 - 0.5)**2 - (q2 - 0.5)**2)
T = T_b + 0.01 * af.exp(-(q1 - 0.5)**2 - (q2 - 0.5)**2)
p1_bulk = 0.01 * af.exp(-(q1 - 0.5)**2 - (q2 - 0.5)**2)
p2_bulk = 0.01 * af.exp(-(q1 - 0.5)**2 - (q2 - 0.5)**2)
p3_bulk = 0.01 * af.exp(-(q1 - 0.5)**2 - (q2 - 0.5)**2)
# Depending on the dimensionality in velocity space, the
# distribution function is assigned accordingly:
if (params.p_dim == 3):
f = rho_b * (m / (2 * np.pi * k * T))**(3 / 2) \
* af.exp(-m * (p1 - p1_bulk)**2 / (2 * k * T)) \
* af.exp(-m * (p2 - p2_bulk)**2 / (2 * k * T)) \
* af.exp(-m * (p3 - p3_bulk)**2 / (2 * k * T))
elif (params.p_dim == 2):
f = rho_b * (m / (2 * np.pi * k * T)) \
* af.exp(-m * (p1 - p1_bulk)**2 / (2 * k * T)) \
* af.exp(-m * (p2 - p2_bulk)**2 / (2 * k * T))
else:
f = rho_b \
* af.sqrt(m / (2 * np.pi * k * T)) \
* af.exp(-m * (p1 - p1_bulk)**2 / (2 * k * T))
af.eval(f)
return (f)
def f0(p1, p2, p3, n, T, p1_bulk, p2_bulk, p3_bulk, params):
"""Return the Local MB distribution."""
m = params.mass_particle
k = params.boltzmann_constant
if (params.p_dim == 3):
f0 = n * (m / (2 * np.pi * k * T))**(3 / 2) \
* af.exp(-m * (p1 - p1_bulk)**2 / (2 * k * T)) \
* af.exp(-m * (p2 - p2_bulk)**2 / (2 * k * T)) \
* af.exp(-m * (p3 - p3_bulk)**2 / (2 * k * T))
elif (params.p_dim == 2):
f0 = n * (m / (2 * np.pi * k * T)) \
* af.exp(-m * (p1 - p1_bulk)**2 / (2 * k * T)) \
* af.exp(-m * (p2 - p2_bulk)**2 / (2 * k * T))
else:
f0 = n * af.sqrt(m / (2 * np.pi * k * T)) \
* af.exp(-m * (p1 - p1_bulk)**2 / (2 * k * T))
af.eval(f0)
return (f0)
def simulateHestonModel(T, N, R, mu, kappa, vBar, sigmaV, rho, x0, v0):
deltaT = T / (float)(N - 1)
x = [
af.constant(x0, R, dtype=af.Dtype.f32),
af.constant(0, R, dtype=af.Dtype.f32)
]
v = [
af.constant(v0, R, dtype=af.Dtype.f32),
af.constant(0, R, dtype=af.Dtype.f32)
]
sqrtDeltaT = math.sqrt(deltaT)
sqrtOneMinusRhoSquare = math.sqrt(1 - rho**2)
m = af.constant(0, 2, dtype=af.Dtype.f32)
m[0] = rho
m[1] = sqrtOneMinusRhoSquare
zeroArray = af.constant(0, R, 1, dtype=af.Dtype.f32)
for t in range(1, N):
tPrevious = (t + 1) % 2
tCurrent = t % 2
dBt = af.randn(R, 2, dtype=af.Dtype.f32) * sqrtDeltaT
vLag = af.maxof(v[tPrevious], zeroArray)
sqrtVLag = af.sqrt(vLag)
x[tCurrent] = x[tPrevious] + (
mu - 0.5 * vLag) * deltaT + sqrtVLag * dBt[:, 0]
v[tCurrent] = vLag + kappa * (vBar - vLag) * deltaT + sigmaV * (
sqrtVLag * af.matmul(dBt, m))
return (x[tCurrent], af.maxof(v[tCurrent], zeroArray))
def simple_arith(verbose=False):
display_func = _util.display_func(verbose)
print_func = _util.print_func(verbose)
a = af.randu(3, 3)
b = af.constant(4, 3, 3)
display_func(a)
display_func(b)
c = a + b
d = a
d += b
display_func(c)
display_func(d)
display_func(a + 2)
display_func(3 + a)
c = a - b
d = a
d -= b
display_func(c)
display_func(d)
display_func(a - 2)
display_func(3 - a)
c = a * b
d = a
d *= b
display_func(c * 2)
display_func(3 * d)
display_func(a * 2)
display_func(3 * a)
c = a / b
d = a
d /= b
display_func(c / 2.0)
display_func(3.0 / d)
display_func(a / 2)
display_func(3 / a)
c = a % b
d = a
d %= b
display_func(c % 2.0)
display_func(3.0 % d)
display_func(a % 2)
display_func(3 % a)
c = a**b
d = a
d **= b
display_func(c**2.0)
display_func(3.0**d)
display_func(a**2)
display_func(3**a)
display_func(a < b)
display_func(a < 0.5)
display_func(0.5 < a)
display_func(a <= b)
display_func(a <= 0.5)
display_func(0.5 <= a)
display_func(a > b)
display_func(a > 0.5)
display_func(0.5 > a)
display_func(a >= b)
display_func(a >= 0.5)
display_func(0.5 >= a)
display_func(a != b)
display_func(a != 0.5)
display_func(0.5 != a)
display_func(a == b)
display_func(a == 0.5)
display_func(0.5 == a)
a = af.randu(3, 3, dtype=af.Dtype.u32)
b = af.constant(4, 3, 3, dtype=af.Dtype.u32)
display_func(a & b)
display_func(a & 2)
c = a
c &= 2
display_func(c)
display_func(a | b)
display_func(a | 2)
c = a
c |= 2
display_func(c)
display_func(a >> b)
display_func(a >> 2)
c = a
c >>= 2
display_func(c)
display_func(a << b)
display_func(a << 2)
c = a
c <<= 2
display_func(c)
display_func(-a)
display_func(+a)
display_func(~a)
display_func(a)
display_func(af.cast(a, af.Dtype.c32))
display_func(af.maxof(a, b))
display_func(af.minof(a, b))
display_func(af.rem(a, b))
a = af.randu(3, 3) - 0.5
b = af.randu(3, 3) - 0.5
display_func(af.abs(a))
display_func(af.arg(a))
display_func(af.sign(a))
display_func(af.round(a))
display_func(af.trunc(a))
display_func(af.floor(a))
display_func(af.ceil(a))
display_func(af.hypot(a, b))
display_func(af.sin(a))
display_func(af.cos(a))
display_func(af.tan(a))
display_func(af.asin(a))
display_func(af.acos(a))
display_func(af.atan(a))
display_func(af.atan2(a, b))
c = af.cplx(a)
d = af.cplx(a, b)
display_func(c)
display_func(d)
display_func(af.real(d))
display_func(af.imag(d))
display_func(af.conjg(d))
display_func(af.sinh(a))
display_func(af.cosh(a))
display_func(af.tanh(a))
display_func(af.asinh(a))
display_func(af.acosh(a))
display_func(af.atanh(a))
a = af.abs(a)
b = af.abs(b)
display_func(af.root(a, b))
display_func(af.pow(a, b))
display_func(af.pow2(a))
display_func(af.sigmoid(a))
display_func(af.exp(a))
display_func(af.expm1(a))
display_func(af.erf(a))
display_func(af.erfc(a))
display_func(af.log(a))
display_func(af.log1p(a))
display_func(af.log10(a))
display_func(af.log2(a))
display_func(af.sqrt(a))
display_func(af.cbrt(a))
a = af.round(5 * af.randu(3, 3) - 1)
b = af.round(5 * af.randu(3, 3) - 1)
display_func(af.factorial(a))
display_func(af.tgamma(a))
display_func(af.lgamma(a))
display_func(af.iszero(a))
display_func(af.isinf(a / b))
display_func(af.isnan(a / a))
a = af.randu(5, 1)
b = af.randu(1, 5)
c = af.broadcast(lambda x, y: x + y, a, b)
display_func(a)
display_func(b)
display_func(c)
@af.broadcast
def test_add(aa, bb):
return aa + bb
display_func(test_add(a, b))
def sqrt(x):
if isinstance(x, afnumpy.ndarray):
s = arrayfire.sqrt(x.d_array)
return afnumpy.ndarray(x.shape, dtype=pu.typemap(s.dtype()), af_array=s)
else:
return numpy.sqrt(x)
def sqrt(arr):
return af.sqrt(arr)
a = af.abs(a) b = af.abs(b) af.display(af.root(a, b)) af.display(af.pow(a, b)) af.display(af.pow2(a)) af.display(af.exp(a)) af.display(af.expm1(a)) af.display(af.erf(a)) af.display(af.erfc(a)) af.display(af.log(a)) af.display(af.log1p(a)) af.display(af.log10(a)) af.display(af.log2(a)) af.display(af.sqrt(a)) af.display(af.cbrt(a)) a = af.round(5 * af.randu(3,3) - 1) b = af.round(5 * af.randu(3,3) - 1) af.display(af.factorial(a)) af.display(af.tgamma(a)) af.display(af.lgamma(a)) af.display(af.iszero(a)) af.display(af.isinf(a/b)) af.display(af.isnan(a/a)) a = af.randu(5, 1) b = af.randu(1, 5) c = af.broadcast(lambda x,y: x+y, a, b)
def simple_arith(verbose = False):
display_func = _util.display_func(verbose)
print_func = _util.print_func(verbose)
a = af.randu(3,3,dtype=af.Dtype.u32)
b = af.constant(4, 3, 3, dtype=af.Dtype.u32)
display_func(a)
display_func(b)
c = a + b
d = a
d += b
display_func(c)
display_func(d)
display_func(a + 2)
display_func(3 + a)
c = a - b
d = a
d -= b
display_func(c)
display_func(d)
display_func(a - 2)
display_func(3 - a)
c = a * b
d = a
d *= b
display_func(c * 2)
display_func(3 * d)
display_func(a * 2)
display_func(3 * a)
c = a / b
d = a
d /= b
display_func(c / 2.0)
display_func(3.0 / d)
display_func(a / 2)
display_func(3 / a)
c = a % b
d = a
d %= b
display_func(c % 2.0)
display_func(3.0 % d)
display_func(a % 2)
display_func(3 % a)
c = a ** b
d = a
d **= b
display_func(c ** 2.0)
display_func(3.0 ** d)
display_func(a ** 2)
display_func(3 ** a)
display_func(a < b)
display_func(a < 0.5)
display_func(0.5 < a)
display_func(a <= b)
display_func(a <= 0.5)
display_func(0.5 <= a)
display_func(a > b)
display_func(a > 0.5)
display_func(0.5 > a)
display_func(a >= b)
display_func(a >= 0.5)
display_func(0.5 >= a)
display_func(a != b)
display_func(a != 0.5)
display_func(0.5 != a)
display_func(a == b)
display_func(a == 0.5)
display_func(0.5 == a)
display_func(a & b)
display_func(a & 2)
c = a
c &= 2
display_func(c)
display_func(a | b)
display_func(a | 2)
c = a
c |= 2
display_func(c)
display_func(a >> b)
display_func(a >> 2)
c = a
c >>= 2
display_func(c)
display_func(a << b)
display_func(a << 2)
c = a
c <<= 2
display_func(c)
display_func(-a)
display_func(+a)
display_func(~a)
display_func(a)
display_func(af.cast(a, af.Dtype.c32))
display_func(af.maxof(a,b))
display_func(af.minof(a,b))
display_func(af.rem(a,b))
a = af.randu(3,3) - 0.5
b = af.randu(3,3) - 0.5
display_func(af.abs(a))
display_func(af.arg(a))
display_func(af.sign(a))
display_func(af.round(a))
display_func(af.trunc(a))
display_func(af.floor(a))
display_func(af.ceil(a))
display_func(af.hypot(a, b))
display_func(af.sin(a))
display_func(af.cos(a))
display_func(af.tan(a))
display_func(af.asin(a))
display_func(af.acos(a))
display_func(af.atan(a))
display_func(af.atan2(a, b))
c = af.cplx(a)
d = af.cplx(a,b)
display_func(c)
display_func(d)
display_func(af.real(d))
display_func(af.imag(d))
display_func(af.conjg(d))
display_func(af.sinh(a))
display_func(af.cosh(a))
display_func(af.tanh(a))
display_func(af.asinh(a))
display_func(af.acosh(a))
display_func(af.atanh(a))
a = af.abs(a)
b = af.abs(b)
display_func(af.root(a, b))
display_func(af.pow(a, b))
display_func(af.pow2(a))
display_func(af.exp(a))
display_func(af.expm1(a))
display_func(af.erf(a))
display_func(af.erfc(a))
display_func(af.log(a))
display_func(af.log1p(a))
display_func(af.log10(a))
display_func(af.log2(a))
display_func(af.sqrt(a))
display_func(af.cbrt(a))
a = af.round(5 * af.randu(3,3) - 1)
b = af.round(5 * af.randu(3,3) - 1)
display_func(af.factorial(a))
display_func(af.tgamma(a))
display_func(af.lgamma(a))
display_func(af.iszero(a))
display_func(af.isinf(a/b))
display_func(af.isnan(a/a))
a = af.randu(5, 1)
b = af.randu(1, 5)
c = af.broadcast(lambda x,y: x+y, a, b)
display_func(a)
display_func(b)
display_func(c)
@af.broadcast
def test_add(aa, bb):
return aa + bb
display_func(test_add(a, b))
a = af.abs(a) b = af.abs(b) af.display(af.root(a, b)) af.display(af.pow(a, b)) af.display(af.pow2(a)) af.display(af.exp(a)) af.display(af.expm1(a)) af.display(af.erf(a)) af.display(af.erfc(a)) af.display(af.log(a)) af.display(af.log1p(a)) af.display(af.log10(a)) af.display(af.log2(a)) af.display(af.sqrt(a)) af.display(af.cbrt(a)) a = af.round(5 * af.randu(3, 3) - 1) b = af.round(5 * af.randu(3, 3) - 1) af.display(af.factorial(a)) af.display(af.tgamma(a)) af.display(af.lgamma(a)) af.display(af.iszero(a)) af.display(af.isinf(a / b)) af.display(af.isnan(a / a)) a = af.randu(5, 1) b = af.randu(1, 5) c = af.broadcast(lambda x, y: x + y, a, b)
