def preserves_hermitian_form(SL2C_matrices):
"""
>>> CC = ComplexField(100)
>>> A = matrix(CC, [[1, 1], [1, 2]]);
>>> B = matrix(CC, [[0, 1], [-1, 0]])
>>> C = matrix(CC, [[CC('I'),0], [0, -CC('I')]])
>>> ans, sig, form = preserves_hermitian_form([A, B])
>>> ans
True
>>> sig
'indefinite'
>>> form.change_ring(ComplexField(10))
[ 0.00 -1.0*I]
[ 1.0*I 0.00]
>>> preserves_hermitian_form([A, B, C])
(False, None, None)
>>> ans, sig, form = preserves_hermitian_form([B, C])
>>> sig
'definite'
"""
M = block_matrix(len(SL2C_matrices), 1, [
left_mult_by_adjoint(A) - right_mult_by_inverse(A)
for A in SL2C_matrices
])
CC = M.base_ring()
mp.prec = CC.prec()
RR = RealField(CC.prec())
epsilon = RR(2)**(-int(0.8 * mp.prec))
U, S, V = mp.svd(sage_matrix_to_mpmath(M))
S = list(mp.chop(S, epsilon))
if mp.zero not in S:
return False, None, None
elif S.count(mp.zero) > 1:
for i, A in enumerate(SL2C_matrices):
for B in SL2C_matrices[i + 1:]:
assert (A * B - B * A).norm() < epsilon
sig = 'indefinite' if any(abs(A.trace()) > 2
for A in SL2C_matrices) else 'both'
return True, sig, None
else:
in_kernel = list(mp.chop(V.H.column(S.index(mp.zero))))
J = mp.matrix([in_kernel[:2], in_kernel[2:]])
iJ = mp.mpc(imag=1) * J
J1, J2 = J + J.H, iJ + iJ.H
J = J1 if mp.norm(J1) >= mp.norm(J2) else J2
J = (1 / mp.sqrt(abs(mp.det(J)))) * J
J = mpmath_matrix_to_sage(J)
assert all((A.conjugate_transpose() * J * A - J).norm() < epsilon
for A in SL2C_matrices)
sig = 'definite' if J.det() > 0 else 'indefinite'
return True, sig, J
def plot_C_pi():
C = []
for m in [0, 1]:
Cm = []
for i in range(0, len(lmx)):
Crow = []
for j in range(0, len(lmy)):
if m == 0:
Crow.append(1)
if m == 1:
Crow.append(
float(
mp.norm(-mp.exp(1j * (mp.atan(
mp.im(eq5.fl(lmx[i][j], lmy[i][j], 0)) /
mp.re(eq5.fl(lmx[i][j], lmx[i][j], 0))))))))
Cm.append(Crow)
C.append(Cm)
for m in range(0, len(C)):
plt.figure()
plt.title("C matrix for band pi, element number " + str(m))
plt.contourf(lmx, lmy, C[m], cmap=plt.get_cmap("coolwarm"))
plt.colorbar()
plt.xlabel(r'$l_x$ $[m^{-1}]$')
plt.ylabel(r'$l_y$ $[m^{-1}]$')
def two_body_reference(self, t1, num_1 = 0, num_2 = 1):
""" reference notations are same as Yoel's notes
using a precision of 64 digits for max accuracy """
mp.dps = 64
self.load_data()
t1 = mp.mpf(t1)
m_1 = mp.mpf(self.planet_lst[num_1].mass)
m_2 = mp.mpf(self.planet_lst[num_2].mass)
x1_0 = mp.matrix(self.planet_lst[num_1].loc)
x2_0 = mp.matrix(self.planet_lst[num_2].loc)
v1_0 = mp.matrix(self.planet_lst[num_1].v)
v2_0 = mp.matrix(self.planet_lst[num_2].v)
M = m_1 + m_2
x_cm = (1/M)*((m_1*x1_0)+(m_2*x2_0))
v_cm = (1/M)*((m_1*v1_0)+(m_2*v2_0))
u1_0 = v1_0 - v_cm
w = x1_0 - x_cm
r_0 = mp.norm(w)
alpha = mp.acos((np.inner(u1_0,w))/(mp.norm(u1_0)*mp.norm(w)))
K = mp.mpf(self.G) * (m_2**3)/(M**2)
u1_0 = mp.norm(u1_0)
L = r_0 * u1_0 * mp.sin(alpha)
cosgamma = ((L**2)/(K*r_0)) - 1
singamma = -((L*u1_0*mp.cos(alpha))/K)
gamma = mp.atan2(singamma, cosgamma)
e = mp.sqrt(((r_0*(u1_0**2)*(mp.sin(alpha)**2)/K)-1)**2 + \
(((r_0**2)*(u1_0**4)*(mp.sin(alpha)**2)*(mp.cos(alpha)**2))/(K**2)) )
r_theta = lambda theta: (L**2)/(K*(1+(e*mp.cos(theta - gamma))))
f = lambda x: r_theta(x)**2/L
curr_theta = 0
max_it = 30
it = 1
converged = 0
while ((it < max_it) & (converged != 1)):
t_val = mp.quad(f,[0,curr_theta]) - t1
dt_val = f(curr_theta)
delta = t_val/dt_val
if (abs(delta)<1.0e-6):
converged = 1
curr_theta -= delta
it += 1
x1_new = mp.matrix([r_theta(curr_theta)*mp.cos(curr_theta), r_theta(curr_theta)*mp.sin(curr_theta)])
# x2_new = -(m_1/m_2)*(x1_new) +(x_cm + v_cm*t1)
x1_new = x1_new + (x_cm + v_cm*t1)
return x1_new
def __init__(self, side, extended, distance):
self.distance = mp.mpf(distance)
rightPointX = mp.mpf('1.41949302') if extended else mp.sqrt(2)
self.side = side # side length of square
self.d = side / mp.cos(mp.pi / 12) # length of left edge
sin60 = mp.sqrt(3) / 2
self.top = matrix([self.d / 2, sin60 * self.d])
self.right = matrix([side * rightPointX, 0])
self.nleft = matrix([[-sin60, 0.5]])
self.ndown = matrix([[0, -1]])
v = self.right - self.top
norm = mp.norm(v)
self.nright = matrix([[-v[1] / norm, v[0] / norm]])
self.offset = self.nright * self.right
def verify(self, x0, x1, y0, y1, s, t, x, y):
"Verify the transformation given by (s, t, x, y) on pentagon."
pentagon = [(x0, 0), (x1, 0), (0, y0), (0, y1), (self.side, self.side)]
# reference point in interior of pentagon
ref = matrix([x0 + x1 + self.side, y0 + y1 + self.side]) / 5
translation = matrix(
[s * ref[0] + t * ref[1] + x, -t * ref[0] + s * ref[1] + y])
# reconstruct rotation angle
norm = mp.sqrt(s * s + t * t)
if t < 0:
angle = -mp.acos(s / norm)
else:
angle = mp.acos(s / norm)
rotation = matrix([[mp.cos(angle), mp.sin(angle)],
[-mp.sin(angle), mp.cos(angle)]])
pentagon = list(map(lambda pt: matrix(pt), pentagon))
ipentagon = list(
map(lambda pt: rotation * (pt - ref) + translation, pentagon))
# Verify that ipentagon is a congruent image of pentagon
eps = mp.mpf(10)**-(mp.dps - 2) # accept error in last two digits
for i in range(5):
for j in range(i + 1, 5):
d1 = mp.norm(pentagon[j] - pentagon[i])
d2 = mp.norm(ipentagon[j] - ipentagon[i])
dd = abs(d1 - d2)
assert dd < eps, (pentagon, dd, eps)
dists = []
for p in ipentagon:
dists.append(self.nleft * p)
dists.append(self.ndown * p)
dists.append(self.nright * p - self.offset)
dist = max(map(lambda m: m[0], dists))
if dist > -self.distance:
sys.stderr.write("Pentagon failing slack test: %d %d %d %d\n" %
(x0, x1, y0, y1))
return False
return True
def solve_multiprecision(M, A, b, tolerance=1e-60):
af = sparse_make_single_precision(M, A)
so = sp.sparse.linalg.factorized(af)
def solve(b):
b = _TO_NUMPY(b).flatten()
x = so(b)
return FROM_NUMPY(x)
"""
def solve(b):
b = _TO_NUMPY(b)
x, e = sp.sparse.linalg.gmres(af, b)
if e != 0:
print("error", e)
raise Exception("gmres error")
return FROM_NUMPY(x)
"""
x = solve(b)
current_prec = mp.prec
try:
guess = np.log2(M) - np.log2(tolerance) + 24
mp.prec = guess
while True:
residual = b - mmul(A, x)
scale = mp.norm(residual, p='inf')
print("residual: %e" % scale)
if scale < tolerance:
return x
compare = solve(residual / scale)
x += scale * compare
amplification = float(mp.norm(compare, p='inf'))
mp.prec = max(mp.prec, guess + np.log2(amplification))
print("precision:", mp.prec)
finally:
mp.prec = current_prec
def nearly_diagonal(A):
assert A.rows == A.cols == 2
a = A[0, 0]
return mp.norm(A - mp.diag([a, a])) < 1000 * mp.eps
def w2s(kx, ky, kz):
return ct.nrm_s * ct.a0**(3 / 2) * (
((ct.a0 * mp.norm([kx, ky, kz])) / ct.hbar)**2 -
(ct.Zeff2s / 2)**2) / (((ct.a0 * mp.norm([kx, ky, kz])) / ct.hbar)**2 +
(ct.Zeff2s / 2)**2)**3
def w2py(kx, ky, kz):
return ct.nrm_py * ct.a0**(5 / 2) * (ky / ct.hbar) / (
(ct.a0 * mp.norm([kx, ky, kz]) / ct.hbar)**2 + (ct.Zeff2pxy / 2)**2)**3
def w2pz(kx, ky, kz):
return ct.nrm_pz * ct.a0**(5 / 2) * (kz / ct.hbar) / (
(ct.a0 * mp.norm([kx, ky, kz]) / ct.hbar)**2 + (ct.Zeff2pz / 2)**2)**3
def speedup_Green_Taylor_Arnoldi_RgNmn_Uconverge(n,
k,
R,
klim=10,
Taylor_tol=1e-8,
invchi=0.1,
Unormtol=1e-4,
veclim=3,
delveclim=2,
plotVectors=False):
#sets up the Arnoldi matrix and associated unit vector lists for any given RgN
#ti = time.time()
kR = mp.mpf(k * R)
pow_jndiv, coe_jndiv, pow_djn, coe_djn, pow_yndiv, coe_yndiv, pow_dyn, coe_dyn = get_Taylor_jndiv_djn_yndiv_dyn(
n, kR, klim, tol=Taylor_tol)
#print(time.time()-ti,'0')
#print(len(pow_jndiv))
pow_jndiv = np.array(pow_jndiv)
pow_djn = np.array(pow_djn)
nfac = mp.sqrt(n * (n + 1))
rmRgN_Bpol = mp.one * np.zeros(pow_jndiv[-1] + 1 - (n - 1))
rmRgN_Ppol = mp.one * np.zeros(pow_jndiv[-1] + 1 - (n - 1))
rmRgN_Bpol[pow_jndiv - (n - 1)] += coe_jndiv
rmRgN_Bpol[pow_djn - (n - 1)] += coe_djn
rmRgN_Ppol[pow_jndiv - (n - 1)] += coe_jndiv
rmRgN_Ppol *= nfac
rmRgN_Bpol = po.Polynomial(rmRgN_Bpol)
rmRgN_Ppol = po.Polynomial(rmRgN_Ppol)
rnImN_Bpol = mp.one * np.zeros(pow_yndiv[-1] + 1 + (n + 2),
dtype=np.complex)
rnImN_Ppol = mp.one * np.zeros(pow_yndiv[-1] + 1 + (n + 2),
dtype=np.complex)
pow_yndiv = np.array(pow_yndiv)
pow_dyn = np.array(pow_dyn)
rnImN_Bpol[(n + 2) + pow_yndiv] += coe_yndiv
rnImN_Bpol[(n + 2) + pow_dyn] += coe_dyn
rnImN_Ppol[(n + 2) + pow_yndiv] += coe_yndiv
rnImN_Ppol *= nfac
rnImN_Bpol = po.Polynomial(rnImN_Bpol)
rnImN_Ppol = po.Polynomial(rnImN_Ppol)
if plotVectors:
plot_rmnNpol(n, rmRgN_Bpol.coef, rmRgN_Ppol.coef, kR * 0.01, kR)
unitrmnBpols = []
unitrmnPpols = []
#print(time.time()-ti,'1')
RgNnorm = mp.sqrt(rmnNnormsqr_Taylor(n, k, R, rmRgN_Bpol, rmRgN_Ppol))
rmnBpol = rmRgN_Bpol / RgNnorm
rmnPpol = rmRgN_Ppol / RgNnorm
unitrmnBpols.append(rmnBpol)
unitrmnPpols.append(rmnPpol)
#print(time.time()-ti,'2')
rmnGBpol, rmnGPpol = rmnGreen_Taylor_Nmn_vec(n, k, R, rmRgN_Bpol,
rmRgN_Ppol, rnImN_Bpol,
rnImN_Ppol, rmnBpol, rmnPpol)
rmnGBpol = rmnGBpol.cutdeg(rmRgN_Bpol.degree())
rmnGPpol = rmnGPpol.cutdeg(rmRgN_Bpol.degree())
#print(time.time()-ti,'3')
prefactpow, Upol = rmnNpol_dot(2 * n - 2, rmnBpol, rmnPpol, rmnGBpol,
rmnGPpol)
Uinv = mp.matrix([[invchi - kR**prefactpow * po.polyval(kR, Upol) / k**3]])
unitrmnBpols.append(rmnGBpol)
unitrmnPpols.append(rmnGPpol) #set up beginning of Arnoldi iteration
#print(time.time()-ti,'4')
b = mp.matrix([mp.one])
prevUnorm = 1 / Uinv[0, 0]
i = 1
while i < veclim:
speedup_Green_Taylor_Arnoldi_step_RgNmn(n,
k,
R,
invchi,
rmRgN_Bpol,
rmRgN_Ppol,
rnImN_Bpol,
rnImN_Ppol,
unitrmnBpols,
unitrmnPpols,
Uinv,
plotVectors=plotVectors)
i += 1
if i == veclim:
#solve for first column of U and see if its norm has converged
b.rows = i
x = mp.lu_solve(Uinv, b)
Unorm = mp.norm(x)
if np.abs(prevUnorm - Unorm) > np.abs(Unorm) * Unormtol:
veclim += delveclim
prevUnorm = Unorm
#else: print(x)
if veclim == 1:
x = mp.one / Uinv[0, 0]
"""
#print(x)
print(mp.norm(x))
plt.figure()
plt.matshow(np.abs(np.array((Uinv**-1).tolist(),dtype=np.complex)))
plt.show()
#EUdUinv = mpmath.eigh(Uinv.transpose_conj()*Uinv, eigvals_only=True)
#print(EUdUinv)
#print(type(EUdUinv))
"""
#print(time.time()-ti,'5')
#returns everything necessary to potentially extend size of Uinv matrix later
return rmRgN_Bpol, rmRgN_Ppol, rnImN_Bpol, rnImN_Ppol, unitrmnBpols, unitrmnPpols, Uinv
def speedup_Green_Taylor_Arnoldi_RgMmn_Uconverge(n,
k,
R,
klim=10,
Taylor_tol=1e-8,
invchi=0.1,
Unormtol=1e-4,
veclim=3,
delveclim=2,
plotVectors=False):
#sets up the Arnoldi matrix and associated unit vector lists for any given RgM
kR = mp.mpf(k * R)
pow_jn, coe_jn, pow_yn, coe_yn = get_Taylor_jn_yn(n, kR, klim, Taylor_tol)
#print(len(pow_jn))
pow_jn = np.array(pow_jn)
coe_jn = np.array(coe_jn)
rmnRgM = mp.one * np.zeros(pow_jn[-1] + 1 - n)
rmnRgM[pow_jn - n] += coe_jn
rmnRgM = po.Polynomial(rmnRgM)
pow_yn = np.array(pow_yn)
coe_yn = np.array(coe_yn)
rnImM = mp.one * np.zeros(pow_yn[-1] + 1 + n + 1)
rnImM[pow_yn + n + 1] += coe_yn
rnImM = po.Polynomial(rnImM)
if plotVectors:
plot_rmnMpol(n, rmnRgM.coef, 0.01 * kR, kR)
unitrmnMpols = []
RgMnorm = mp.sqrt(rmnMnormsqr_Taylor(n, k, R, rmnRgM))
rmnMpol = rmnRgM / RgMnorm
unitrmnMpols.append(rmnMpol)
rmnGMpol = rmnGreen_Taylor_Mmn_vec(n, k, R, rmnRgM, rnImM, rmnMpol)
rmnGMpol = rmnGMpol.cutdeg(rmnRgM.degree())
prefactpow, Upol = rmnMpol_dot(2 * n, rmnMpol, rmnGMpol)
Uinv = mp.matrix([[invchi - kR**prefactpow * po.polyval(kR, Upol) / k**3]])
unitrmnMpols.append(rmnGMpol) #set up beginning of Arnoldi iteration
b = mp.matrix([mp.one])
prevUnorm = 1 / Uinv[0, 0]
i = 1
while i < veclim:
speedup_Green_Taylor_Arnoldi_step_RgMmn(n,
k,
R,
invchi,
rmnRgM,
rnImM,
unitrmnMpols,
Uinv,
plotVectors=plotVectors)
i += 1
if i == veclim:
#solve for first column of U and see if its norm has converged
b.rows = i
x = mp.lu_solve(Uinv, b)
Unorm = mp.norm(x)
if np.abs(prevUnorm - Unorm) > np.abs(Unorm) * Unormtol:
veclim += delveclim
prevUnorm = Unorm
#else: print(x)
if veclim == 1:
x = mp.one / Uinv[0, 0]
"""
print(x)
print(mp.norm(x))
plt.figure()
plt.matshow(np.abs(np.array((Uinv**-1).tolist(),dtype=np.complex)))
plt.show()
#EUdUinv = mpmath.eigh(Uinv.transpose_conj()*Uinv, eigvals_only=True)
#print(EUdUinv)
"""
#returns everything necessary to potentially extend size of Uinv matrix later
return rmnRgM, rnImM, unitrmnMpols, Uinv
def zdipole_field_xy2D_periodic_array(k0,
L,
xd,
yd,
zd,
xp,
yp,
zp,
sumtol=1e-12):
#field at coord (xp,yp,zp) of z-polarized dipoles in a square array in the z=czd plane
#generated by summing over discrete k-vectors; in evanescent region the summand decreases exponentially with increasing kp and abskz
#it seems that directly summing over individual dipole fields leads to convergence issues due to relatively slow (polynomial) decay of dipole fields
#order of summation in k-space: concentric squares, 0th square the origin, 1st square infinity-norm radius 1, 2nd square infinity-norm radius 2...
#since we are using the angular representation for consistency we insist that zp>zd
"""
if not (xp<L/2 and xp>-L/2 and yp<L/2 and yp>-L/2):
return 'target point out of Brillouin zone'
"""
if zp <= zd:
return 'need zp>zd'
field = mp.zeros(3, 1)
oldfield = mp.zeros(3, 1)
deltak = 2 * mp.pi / L
deltax = xp - xd
deltay = yp - yd
deltaz = zp - zd
i = 1 #label of which square we are on
prefac = -1j / (2 * L**2 * k0**2)
while True: #termination condition in loop
#sum over the i-th square
lkx = -i * deltak
rkx = i * deltak
for iy in range(-i, i + 1):
ky = iy * deltak
kpsqr = lkx**2 + ky**2
kz = mp.sqrt(k0**2 - kpsqr)
lphasefac = mp.expj(lkx * deltax + ky * deltay + kz * deltaz)
rphasefac = mp.expj(rkx * deltax + ky * deltay + kz * deltaz)
field[0] += prefac * (lkx * lphasefac + rkx * rphasefac)
field[1] += prefac * (lphasefac + rphasefac) * ky
field[2] += prefac * (lphasefac + rphasefac) * (-kpsqr / kz)
bky = -i * deltak
uky = i * deltak
for ix in range(-i + 1, i):
kx = ix * deltak
kpsqr = kx**2 + bky**2
kz = mp.sqrt(k0**2 - kpsqr)
bphasefac = mp.expj(kx * deltax + bky * deltay + kz * deltaz)
uphasefac = mp.expj(kx * deltax + uky * deltay + kz * deltaz)
field[0] += prefac * (bphasefac + uphasefac) * kx
field[1] += prefac * (bphasefac * bky + uphasefac * uky)
field[2] += prefac * (bphasefac + uphasefac) * (-kpsqr / kz)
if mp.norm(field - oldfield) < mp.norm(field) * sumtol:
break
print('i', i)
#mp.nprint(field)
oldfield = field.copy()
i += 1
return field
def shell_Green_grid_Arnoldi_RgandImMmn_Uconverge_mp(n,k,R1,R2, invchi, gridpts=1000, Unormtol=1e-10, veclim=3, delveclim=2, maxveclim=40, plotVectors=False):
np.seterr(over='raise',under='raise',invalid='raise')
#for high angular momentum number could have floating point issues; in this case, raise error. Outer method will catch the error and use the mpmath version instead
rgrid = np.linspace(R1,R2,gridpts)
rsqrgrid = rgrid**2
rdiffgrid = np.diff(rgrid)
"""
RgMgrid = sp.spherical_jn(n, k*rgrid) #the argument for radial part of spherical waves is kr
ImMgrid = sp.spherical_yn(n, k*rgrid)
RgMgrid = RgMgrid.astype(np.complex)
ImMgrid = ImMgrid.astype(np.complex)
RgMgrid = complex_to_mp(RgMgrid)
ImMgrid = complex_to_mp(ImMgrid)
"""
RgMgrid = mp_vec_spherical_jn(n, k*rgrid)
ImMgrid = mp_vec_spherical_yn(n, k*rgrid)
vecRgMgrid = RgMgrid / mp.sqrt(rgrid_Mmn_normsqr(RgMgrid, rsqrgrid,rdiffgrid))
vecImMgrid = ImMgrid - rgrid_Mmn_vdot(vecRgMgrid, ImMgrid, rsqrgrid,rdiffgrid)*vecRgMgrid
vecImMgrid /= mp.sqrt(rgrid_Mmn_normsqr(vecImMgrid,rsqrgrid,rdiffgrid))
if plotVectors:
rgrid_Mmn_plot(vecRgMgrid,rgrid)
rgrid_Mmn_plot(vecImMgrid,rgrid)
unitMvecs = [vecRgMgrid,vecImMgrid]
GvecRgMgrid = shell_Green_grid_Mmn_vec_mp(n,k, rsqrgrid,rdiffgrid, RgMgrid,ImMgrid, vecRgMgrid)
GvecImMgrid = shell_Green_grid_Mmn_vec_mp(n,k, rsqrgrid,rdiffgrid, RgMgrid,ImMgrid, vecImMgrid)
Gmat = mp.zeros(2,2)
Gmat[0,0] = rgrid_Mmn_vdot(vecRgMgrid, GvecRgMgrid, rsqrgrid,rdiffgrid)
Gmat[0,1] = rgrid_Mmn_vdot(vecRgMgrid, GvecImMgrid, rsqrgrid,rdiffgrid)
Gmat[1,0] = Gmat[0,1]
Gmat[1,1] = rgrid_Mmn_vdot(vecImMgrid,GvecImMgrid, rsqrgrid,rdiffgrid)
Uinv = mp.eye(2)*invchi-Gmat
unitMvecs.append(GvecRgMgrid)
unitMvecs.append(GvecImMgrid) #append unorthogonalized, unnormalized Arnoldi vector for further iterations
b = mp.matrix([mp.one])
prevUnorm = 1 / Uinv[0,0]
i=2
while i<veclim:
Gmat = shell_Green_grid_Arnoldi_RgandImMmn_step_mp(n,k,invchi, rgrid,rsqrgrid,rdiffgrid, RgMgrid, ImMgrid, unitMvecs, Gmat, plotVectors=plotVectors)
i += 1
print(i)
if i==maxveclim:
break
if i==veclim:
#solve for first column of U and see if its norm has converged
Uinv = mp.eye(Gmat.rows)*invchi-Gmat
b.rows = i
x = mp.lu_solve(Uinv, b)
Unorm = mp.norm(x)
print('Unorm', Unorm, flush=True)
if np.abs(prevUnorm-Unorm) > np.abs(Unorm)*Unormtol:
veclim += delveclim
prevUnorm = Unorm
return rgrid,rsqrgrid,rdiffgrid, RgMgrid, ImMgrid, unitMvecs, Uinv, Gmat