def print_fun_results(x):
res1 = float(rf(x))
res2 = float(fp(x))
print res1
print res2
print res1 - res2
print fdistan_two(res1, res2)
print special.airy(x)
print special.ai_zeros(1)
def test_trangle_well(self):
F = 2e-4
xmax = 1E3
mass = 0.067 # Typical parameters for GaAs
x = np.linspace(0, xmax, 200)
step = x[1] - x[0]
V = F * (xmax - x)
Es = np.linspace(0, 0.2, 300)
EigenEs = cSimpleSolve1D(step, Es, V, mass)
psis = cSimpleFillPsi(step, EigenEs, V, mass)
# Theoretical result
nmax = 8
an = ai_zeros(nmax)[0]
EigenEs_th = -(hbar**2 * F**2 /
(2 * m_e * mass * e * ANG**2))**(1 / 3) * an
psis_th = np.array([
airy((2 * mass * m_e * e * ANG**2 * F / hbar**2)**(1 / 3) *
(x - E / F))[0] for E in EigenEs_th
])
psis_th /= (np.linalg.norm(psis_th, axis=1) * np.sqrt(step))[:, None]
np.testing.assert_array_almost_equal(EigenEs[:nmax],
EigenEs_th,
decimal=E_dec,
verbose=True)
np.testing.assert_array_almost_equal(psis[:nmax, ::-1],
psis_th,
decimal=wf_dec,
verbose=True)
def triangle_well(F, xmax=1E3):
x = np.linspace(0, xmax, 5000)
step = x[1] - x[0]
V = F * (xmax - x)
mass = 0.067
Es = np.linspace(0, 0.15, 100)
EigenEs = cSimpleSolve1D(step, Es, V, mass)
psis = cSimpleFillPsi(step, EigenEs, V, mass)
# an = np.arange(0, 8) + 0.75
# EigenEs_th = (hbar**2/(2*m_e*mass*e*ANG**2)*(3*pi*F*an/2)**2)**(1/3)
# Above is approximate Airy zeros result
from scipy.special import airy, ai_zeros
an = ai_zeros(8)[0]
EigenEs_th = -(hbar**2 * F**2 /
(2 * m_e * mass * e * ANG**2))**(1 / 3) * an
psis_th = np.array([
airy((2 * mass * m_e * e * ANG**2 * F / hbar**2)**(1 / 3) *
(x - E / F))[0] for E in EigenEs_th
])
psis_th /= (np.linalg.norm(psis_th, axis=1) * sqrt(step))[:, None]
V = np.ascontiguousarray(V[::-1])
psis = np.ascontiguousarray(psis[:, ::-1])
print("Eigen Energy: ", EigenEs)
print("Thoery: ", EigenEs_th)
print("Diff: ", EigenEs - EigenEs_th)
plot(x, V)
scale = 0.15
for n in range(0, EigenEs.size):
plot(x, EigenEs[n] + psis[n, :] * scale)
for n in range(0, EigenEs_th.size):
plot(x, EigenEs_th[n] + psis_th[n, :] * scale, '--')
return x, V, EigenEs, psis, EigenEs_th, psis_th
def testAiry(self):
def fAiry(x):
f = special.airy(x)
return f[0]
f = fAiry
solver = newton.Newton(f, tol=1.e-15, maxiter=100)
x = solver.solve(-1.5)
x0 = special.ai_zeros(1)
x0 = x0[0]
self.assertAlmostEqual(x, x0)
def time_ai_zeros(self):
ai_zeros(100000)
def time_ai_zeros(self):
ai_zeros(100000)
from scipy import special as sp print sp.ai_zeros(3)
def calc_zeta(n):
"""Calculate the nth zero of the Airy function."""
return special.ai_zeros(n)[0][-1]
rectgauss = sym.exp(-u_x**4-u_y**4)
stgauss = gauss*(1. - A*(sym.sin(B*(u_x))+sym.sin(B*(u_y - 2*pi*0.3))))
asymgauss = sym.exp(-u_x**2-u_y**4)
supergauss2 = sym.exp(-(u_x**2+u_y**2)**2)
supergauss3 = sym.exp(-(u_x**2+u_y**2)**3)
superlorentz = 1./((u_x**2 + u_y**2)**2+1.)
lensf = gauss
lensg = np.array([sym.diff(lensf, u_x), sym.diff(lensf, u_y)])
lensh = np.array([sym.diff(lensf, u_x, u_x), sym.diff(lensf, u_y, u_y), sym.diff(lensf, u_x, u_y)])
lensgh = np.array([sym.diff(lensf, u_x, u_x, u_x), sym.diff(lensf, u_x, u_x, u_y), sym.diff(lensf, u_x, u_y, u_y), sym.diff(lensf, u_y, u_y, u_y)])
lensfun = sym.lambdify([u_x, u_y], lensf, "numpy")
lensg = sym.lambdify([u_x, u_y], lensg, "numpy")
lensh = sym.lambdify([u_x, u_y], lensh, "numpy")
lensgh = sym.lambdify([u_x, u_y], lensgh, "numpy")
airsqrenv = interp1d(ai_zeros(100)[1], airy(ai_zeros(100)[1])[0]**2/2., kind = 'cubic', fill_value = 'extrapolate')
@jit(nopython=True)
def alpha(dso, dsl, f, dm):
""" Returns alpha coefficient. """
dlo = dso - dsl
return -c**2*re*dsl*dlo*dm/(2*pi*f**2*dso)
@jit(nopython=True)
def rFsqr(dso, dsl, f):
""" Returns the square of the Fresnel scale. """
dlo = dso - dsl
return c*dsl*dlo/(2*pi*f*dso)
@jit(nopython=True)
def lensc(dm, f):
def exact(n, l, k):
zn = -sp.ai_zeros(4)[0]
return 2. - 4. * epsilon * np.cos(2. * np.pi * k / l) + 4. * epsilon * (
(0.5 * np.cos(2. * np.pi * k / l))**(1. / 3.)) * h**(2. / 3.) * zn[n]
from scipy.special import eval_hermite, gamma, ai_zeros
def V(x): # potential
return beta * abs(x)
def uVu(x): # integrand $u_m V u_n$
y, c = x / a0, a0 * np.sqrt(
np.pi * gamma(m + 1) * gamma(n + 1) * 2**(m + n))
return V(x) * eval_hermite(m, y) * eval_hermite(n, y) * np.exp(-y * y) / c
beta, omega, N = 1., 1., 20 # pot slope, nat freq, num. basis states
a0, x0 = 1 / np.sqrt(omega), 1 / (2 * beta)**(1. / 3) # length scale, x0
m, Vm = np.arange(N - 2), np.zeros((N, N))
mm = np.sqrt((m + 1) * (m + 2)) # calc T matrix
Tm = np.diag(np.arange(1, N + N, 2, float)) # diagonal
Tm -= np.diag(mm, 2) + np.diag(mm, -2) # off diagonal by +/-2
for m in range(N): # calc V matrix
for n in range(m, N, 2): # m+n is even every 2 steps
Vm[m, n] = 2 * itg.gauss(uVu, 0., 10 * a0) # use symm.
if (m != n): Vm[n, m] = Vm[m, n]
Tm = Tm * omega / 4.
E, u = eigsh(Tm + Vm, 6, which='SA') # get lowest 6 states
zn, zpn, zbn, zbpn = ai_zeros(3) # find exact values
Ex = -beta * x0 * np.insert(zpn, range(1, 4), zn) # combine even/odd
print(E, E / Ex)
field = amp * np.exp(1j * (phase + pshift))
dynspec = np.real(np.multiply(field, np.conj(field)))
return rays, amp, phase, field, dynspec, pshift
# We are going to solve the lens equation to find the mapping between the
# u' and u plane numerically using a root finding algorithm.
# For now, however, this won't be necessary - but we'll set up the tools for later (3.2.21).
# Compute 100 zeros and values of the Airy function Ai and its derivative
# where index 1 for ai_zeros() returns the first 100 zeros of Ai’(x) and
# index 0 for airy() returns Ai'(x).
# Note for later: A caustic in which two images merge corresponds to a
# fold catastrophe, and close to a fold the field follows an Airy function pattern.
airyzeros = ai_zeros(100)[1]
airyfunc = airy(airyzeros)[0]**2 / 2.
airsqrenv = interp1d(airyzeros,
airyfunc,
kind='cubic',
fill_value='extrapolate')
# Define screen paramters (Gaussian & Kolmogorov)
nscreen = 100
N = 10
sigma = 2
iscreen = 1 #which wavefront to plot
dso = 1. * kpc * pctocm #distance from source to observer
dsl = 1. * kpc * pctocm / nscreen #distance from source to screen
dm = -1e-5 * pctocm #dispersion measure
#ax, ay = 0.04*autocm, 0.04*autocm #screen width (x,y)
import numpy as np, integral as itg
from scipy.sparse.linalg import eigsh
from scipy.special import eval_hermite, gamma, ai_zeros
def V(x): # potential
return beta*abs(x)
def uVu(x): # integrand $u_m V u_n$
y, c = x/a0, a0*np.sqrt(np.pi*gamma(m+1)*gamma(n+1)*2**(m+n))
return V(x)*eval_hermite(m,y)*eval_hermite(n,y)*np.exp(-y*y)/c
beta, omega, N = 1., 1., 20 # pot slope, nat freq, num. basis states
a0, x0 = 1/np.sqrt(omega), 1/(2*beta)**(1./3) # length scale, x0
m, Vm = np.arange(N-2), np.zeros((N,N))
mm = np.sqrt((m+1)*(m+2)) # calc T matrix
Tm = np.diag(np.arange(1, N+N, 2,float)) # diagonal
Tm -= np.diag(mm, 2) + np.diag(mm, -2) # off diagonal by +/-2
for m in range(N): # calc V matrix
for n in range(m, N, 2): # m+n is even every 2 steps
Vm[m, n] = 2*itg.gauss(uVu, 0., 10*a0) # use symm.
if (m != n): Vm[n,m] = Vm[m,n]
Tm = Tm*omega/4.
E, u = eigsh(Tm + Vm, 6, which='SA') # get lowest 6 states
zn, zpn, zbn, zbpn = ai_zeros(3) # find exact values
Ex = - beta*x0*np.insert(zpn, range(1, 4), zn) # combine even/odd
print (E, E/Ex)