def __hyperplane_eval_iso_2p5D(I, x, r, k0, k1, w0, arr):
''' k[i0,i1,i2] = k0[i1]*w0[i0] + k1[i2]*w1[i0] '''
i0, i1, i2 = cuda.grid(3)
tmp = 0
K0 = k0[i1] * w0[i0, 0]
K1 = k0[i1] * w0[i0, 1]
K2 = k1[i2]
for ii in range(x.shape[0]):
norm = (K0 * K0 + K1 * K1 + K2 * K2) / (r[ii, 0] * r[ii, 0])
if norm < 30:
IP = x[ii, 0] * K0 + x[ii, 1] * K1 + x[ii, 2] * K2
tmp += I[ii] * c_exp(-.5 * norm) * c_exp(complex(0, -IP))
arr[i0, i1, i2] = tmp
def __grid_eval_iso(I, x, r, k0, k1, k2, arr):
'''
I*exp(-|rk|^2/2)*exp(-(x\cdot k)i)
'''
i0, i1, i2 = cuda.grid(3)
tmp = 0
for ii in range(x.shape[0]):
norm = (k0[i0] * k0[i0] + k1[i1] * k1[i1] +
k2[i2] * k2[i2]) / (r[ii, 0] * r[ii, 0])
if norm < 30:
IP = x[ii, 0] * k0[i0] + x[ii, 1] * k1[i1] + x[ii, 2] * k2[i2]
tmp += I[ii] * c_exp(-.5 * norm) * c_exp(complex(0, -IP))
arr[i0, i1, i2] = tmp
def __grid_eval_aniso(I, x, r, k0, k1, k2, arr):
'''
I*exp(-|rk|^2/2)*exp(-(x\cdot k)i)
'''
i0, i1, i2 = cuda.grid(3)
tmp = 0
for ii in range(x.shape[0]):
rk0 = r[ii, 0] * k0[i0]
rk1 = r[ii, 3] * k0[i0] + r[ii, 1] * k1[i1]
rk2 = r[ii, 5] * k0[i0] + r[ii, 4] * k1[i1] + r[ii, 2] * k2[i2]
norm = rk0 * rk0 + rk1 * rk1 + rk2 * rk2
if norm < 30:
IP = x[ii, 0] * k0[i0] + x[ii, 1] * k1[i1] + x[ii, 2] * k2[i2]
tmp += I[ii] * c_exp(-.5 * norm) * c_exp(complex(0, -IP))
arr[i0, i1, i2] = tmp
def __hyperplane_eval_aniso_2p5D(I, x, r, k0, k1, w0, arr):
''' k[i0,i1,i2] = k0[i1]*w0[i0] + k1[i2]*w1[i0] '''
i0, i1, i2 = cuda.grid(3)
tmp = 0
K0 = k0[i1] * w0[i0, 0]
K1 = k0[i1] * w0[i0, 1]
K2 = k1[i2]
for ii in range(x.shape[0]):
rk0 = r[ii, 0] * K0
rk1 = r[ii, 3] * K0 + r[ii, 1] * K1
rk2 = r[ii, 5] * K0 + r[ii, 4] * K1 + r[ii, 2] * K2
norm = rk0 * rk0 + rk1 * rk1 + rk2 * rk2
if norm < 30:
IP = x[ii, 0] * K0 + x[ii, 1] * K1 + x[ii, 2] * K2
tmp += I[ii] * c_exp(-.5 * norm) * c_exp(complex(0, -IP))
arr[i0, i1, i2] = tmp
def q4():
"For question 4 of the assignment."
domain = np.linspace(-.5, .5, 128)
f1 = lambda x: (1-0.9*c_exp(-24j*pi*x)-0.5*c_exp(-2j*pi*x)+\
0.045*c_exp(-26j*pi*x))
f2 = lambda x: (1 - 0.5 * c_exp(-2j * pi * x))
f3 = lambda x: (1 - 0.9 * c_exp(-4j * pi * x))
g1 = lambda y: 1 / (np.absolute(f1(y))**2)
g2 = lambda y: 1 / (np.absolute(f2(y))**2)
g3 = lambda y: 1 / (np.absolute(f3(y))**2)
pts1 = [g1(i) for i in domain]
pts2 = [g2(i) for i in domain]
pts3 = [g3(i) for i in domain]
plt.plot(domain, pts1)
plt.title("Spectral Density Q4PB")
plt.show()
plt.plot(domain, pts2)
plt.title("Spectral Density Q4PC, Part 1")
plt.show()
plt.plot(domain, pts3)
plt.title("Spectral Density Q4PC, Part 2")
plt.show()
def q6():
"For question 6 of the assignment."
domain = np.linspace(-.5, .5, 128)
f1 = lambda x: (1 - 0.99 * c_exp(-6j * pi * x))
g1 = lambda y: 1 / (np.absolute(f1(y))**2)
pts1 = [g1(i) for i in domain]
plt.plot(domain, pts1)
plt.title("Spectral Density Q6PA")
plt.show()
helper_func = lambda x: (2 * (sin(2 * pi * x) * (1 / tan(pi * x))) - 1) / 3
trans_func = lambda y: helper_func(y)**2
f2 = lambda x: g1(x) * trans_func(x)
pts2 = [f2(i) for i in domain]
plt.plot(domain, pts2)
plt.title("Spectral Density Q6PD")
plt.show()
df = pd.DataFrame(pts1)
new_data = pd.rolling_mean(df, window=3)
plt.plot(domain, new_data)
plt.show()
def __derivs_iso_aux(I, x, r, K, g, dg, ddg):
'''
No mixed terms so we have:
g = I*exp(-|k/r|^2/2)*exp(-(x\cdot k)i)
= I*G
dg/dI = G
dg/dx_j = -ik_jg
dg/dr = +(k/r^2)(k/r)g
= |k/r|^2g/r
ddg/dII = 0
ddg/dxx = -kk g
ddg/drr = |k|^2g(|k|^2/r^6 - 3/r^4)
= (|k/r|^2-3)|k/r|^2g/r^2
ddg/dIx_j = -ik_jG
ddg/dIr = |k/r|^2G/r
ddg/dxr = -ik (dg/dr)
'''
rk = cuda.local.array((3, ), f4)
rk[0] = K[0] / r
rk[1] = K[1] / r
rk[2] = K[2] / r
N = rk[0] * rk[0] + rk[1] * rk[1] + rk[2] * rk[2]
if N > 60:
g[0] = 0
for i in range(10):
dg[i] = 0
for j in range(10):
ddg[i, j] = 0
return
G = c_exp(complex(-.5 * N, -(x[0] * K[0] + x[1] * K[1] + x[2] * K[2])))
g[0] = I * G
# d/dI
dg[0] = G
# d^2/dI^2
ddg[0, 0] = 0
# d^2/dx^2
for i in range(3):
for j in range(i, 3):
ddg[1 + i, 1 + j] = -K[i] * K[j] * g[0]
# d^2/dr^2
ddg[4, 4] = (N - 3) * N * g[0] / (r * r)
# d^2/dIdx
for i in range(3):
ddg[0, 1 + i] = complex(0, -K[i]) * G
# d/dx
for i in range(3):
dg[1 + i] = I * ddg[0, 1 + i]
# d^2/dIdr
ddg[0, 4] = N * G / r
# d/dr
dg[4] = ddg[0, 4] * I
# d^2/dxdr
for i in range(3):
ddg[1 + i, 4] = complex(0, -K[i]) * dg[4]
# Symetrise
for i in range(5):
for j in range(i):
ddg[i, j] = ddg[j, i]
def __derivs_aniso_aux(I, x, r, K, g, dg, ddg):
'''
No mixed terms so we have:
g = I*exp(-|rk|^2/2)*exp(-(x\cdot k)i)
= I*G
dg/dI = G
dg/dx_j = -ik_jg
dg/dr_{j0,j1} = -rkg (drk/dr_{j0,j1})
= -(rk)_{j0}k_{j1}g
ddg/dII = 0
ddg/dxx = -kk g
ddg/dr_{j0,j1}r_{j2,j3} = (rk)_{j0}(rk)_{j2}k_{j1}k_{j3} g - k_{j1}k_{j3}g \delta_{j0==j2}
ddg/dIx_j = -ik_jG
ddg/dIr_j = -(rk)_{j0}k_{j1}G
ddg/dxr = -ik (dg/dr)
'''
rk = cuda.local.array((3, ), f4)
ind0 = cuda.local.array((6, ), i4)
ind1 = cuda.local.array((6, ), i4)
ind0[0], ind0[1], ind0[2] = 0, 1, 2
ind0[3], ind0[4], ind0[5] = 1, 2, 2
ind1[0], ind1[1], ind1[2] = 0, 1, 2
ind1[3], ind1[4], ind1[5] = 0, 1, 0
rk[0] = r[0] * K[0]
rk[1] = r[3] * K[0] + r[1] * K[1]
rk[2] = r[5] * K[0] + r[4] * K[1] + r[2] * K[2]
N = -.5 * (rk[0] * rk[0] + rk[1] * rk[1] + rk[2] * rk[2])
if N < -30:
g[0] = 0
for i in range(10):
dg[i] = 0
for j in range(10):
ddg[i, j] = 0
return
G = c_exp(complex(N, -(x[0] * K[0] + x[1] * K[1] + x[2] * K[2])))
g[0] = I * G
# d/dI
dg[0] = G
# d^2/dI^2
ddg[0, 0] = 0
# d^2/dx^2
for i in range(3):
for j in range(i, 3):
ddg[1 + i, 1 + j] = -K[i] * K[j] * g[0]
# d^2/dr^2
for i in range(6):
i0, i1 = ind0[i], ind1[i]
ddg[4 + i,
4 + i] = g[0] * (rk[i0] * K[i1] * rk[i0] * K[i1] - K[i1] * K[i1])
for j in range(i + 1, 6):
j0, j1 = ind0[j], ind1[j]
if i0 == j0:
ddg[4 + i, 4 + j] = g[0] * (rk[i0] * K[i1] * rk[j0] * K[j1] -
K[i1] * K[j1])
else:
ddg[4 + i, 4 + j] = g[0] * rk[i0] * K[i1] * rk[j0] * K[j1]
# d^2/dIdx
for i in range(3):
ddg[0, 1 + i] = complex(0, -K[i]) * G
# d/dx
for i in range(3):
dg[1 + i] = I * ddg[0, 1 + i]
# d^2/dIdr
for i in range(6):
i0, i1 = ind0[i], ind1[i]
ddg[0, 4 + i] = -rk[i0] * K[i1] * G
# d/dr
for i in range(6):
dg[4 + i] = ddg[0, 4 + i] * I
# d^2/dxdr
for i in range(3):
for j in range(6):
ddg[1 + i, 4 + j] = complex(0, -K[i]) * dg[4 + j]
# Symetrise
for i in range(10):
for j in range(i):
ddg[i, j] = ddg[j, i]
