def test_trivial_cholesky(self):
# Known starting point of SI_nn = <psit_m|S+alpha*I|psit_n>
I_nn = np.eye(*self.S0_nn.shape)
alpha = 1e-3 # shift eigenvalues away from zero
SI_nn = self.S0_nn + alpha * I_nn
# Try Cholesky decomposition SI_nn = L_nn * L_nn^dag
L_nn = np.linalg.cholesky(SI_nn)
# |psit_n> -> C_nn |psit_n> , C_nn^(-1) = L_nn^dag
# <psit_m|SI|psit_n> -> <psit_m|C_nn^dag SI C_nn|psit_n> = diag(W_n)
C_nn = np.linalg.inv(L_nn.T.conj())
# Set up Hermitian overlap operator:
S = lambda x: x
dS = lambda a, P_ni: np.dot(P_ni, self.setups[a].dO_ii)
nblocks = self.get_optimal_number_of_blocks(self.blocking)
overlap = MatrixOperator(self.ksl, nblocks, self.async, True)
self.psit_nG = overlap.matrix_multiply(C_nn.T.copy(), self.psit_nG, self.P_ani)
D_nn = overlap.calculate_matrix_elements(self.psit_nG, \
self.P_ani, S, dS).T.copy() # transpose to get <psit_m|A|psit_n>
tri2full(D_nn, 'U') # upper to lower...
if self.bd.comm.rank == 0:
self.gd.comm.broadcast(D_nn, 0)
self.bd.comm.broadcast(D_nn, 0)
if memstats:
self.mem_test = record_memory()
# D_nn = C_nn^dag * S_nn * C_nn = I_nn - alpha * C_nn^dag * C_nn
D0_nn = I_nn - alpha * np.dot(C_nn.T.conj(), C_nn)
self.check_and_plot(D_nn, D0_nn, 6, 'trivial,cholesky') #XXX precision
def test_multiply_randomized(self):
# Known starting point of S_nn = <psit_m|S|psit_n>
S_nn = self.S0_nn
if self.dtype == complex:
C_nn = np.random.uniform(size=self.nbands**2) * \
np.exp(1j*np.random.uniform(0,2*np.pi,size=self.nbands**2))
else:
C_nn = np.random.normal(size=self.nbands**2)
C_nn = C_nn.reshape((self.nbands,self.nbands)) / np.linalg.norm(C_nn,2)
world.broadcast(C_nn, 0)
# Set up Hermitian overlap operator:
S = lambda x: x
dS = lambda a, P_ni: np.dot(P_ni, self.setups[a].dO_ii)
nblocks = self.get_optimal_number_of_blocks(self.blocking)
overlap = MatrixOperator(self.ksl, nblocks, self.async, True)
self.psit_nG = overlap.matrix_multiply(C_nn.T.copy(), self.psit_nG, self.P_ani)
D_nn = overlap.calculate_matrix_elements(self.psit_nG, \
self.P_ani, S, dS).T.copy() # transpose to get <psit_m|A|psit_n>
tri2full(D_nn, 'U') # upper to lower...
if self.bd.comm.rank == 0:
self.gd.comm.broadcast(D_nn, 0)
self.bd.comm.broadcast(D_nn, 0)
if memstats:
self.mem_test = record_memory()
# D_nn = C_nn^dag * S_nn * C_nn
D0_nn = np.dot(C_nn.T.conj(), np.dot(S_nn, C_nn))
self.check_and_plot(D_nn, D0_nn, 9, 'multiply,randomized')
def test_trivial_diagonalize(self):
# Known starting point of S_nn = <psit_m|S|psit_n>
S_nn = self.S0_nn
# Eigenvector decomposition S_nn = V_nn * W_nn * V_nn^dag
# Utilize the fact that they are analytically known (cf. Maple)
band_indices = np.arange(self.nbands)
V_nn = np.eye(self.nbands).astype(self.dtype)
if self.dtype == complex:
V_nn[1:,1] = np.conj(self.gamma)**band_indices[1:] * band_indices[1:]**0.5
V_nn[1,2:] = -self.gamma**band_indices[1:-1] * band_indices[2:]**0.5
else:
V_nn[2:,1] = band_indices[2:]**0.5
V_nn[1,2:] = -band_indices[2:]**0.5
W_n = np.zeros(self.nbands).astype(self.dtype)
W_n[1] = (1. + self.Qtotal) * self.nbands * (self.nbands - 1) / 2.
# Find the inverse basis
Vinv_nn = np.linalg.inv(V_nn)
# Test analytical eigenvectors for consistency against analytical S_nn
D_nn = np.dot(Vinv_nn, np.dot(S_nn, V_nn))
self.assertAlmostEqual(np.abs(D_nn.diagonal()-W_n).max(), 0, 8)
self.assertAlmostEqual(np.abs(np.tril(D_nn, -1)).max(), 0, 4)
self.assertAlmostEqual(np.abs(np.triu(D_nn, 1)).max(), 0, 4)
del Vinv_nn, D_nn
# Perform Gram Schmidt orthonormalization for diagonalization
# |psit_n> -> C_nn |psit_n>, using orthonormalized basis Q_nn
# <psit_m|S|psit_n> -> <psit_m|C_nn^dag S C_nn|psit_n> = diag(W_n)
# using S_nn = V_nn * W_nn * V_nn^(-1) = Q_nn * W_nn * Q_nn^dag
C_nn = V_nn.copy()
gram_schmidt(C_nn)
self.assertAlmostEqual(np.abs(np.dot(C_nn.T.conj(), C_nn) \
- np.eye(self.nbands)).max(), 0, 6)
# Set up Hermitian overlap operator:
S = lambda x: x
dS = lambda a, P_ni: np.dot(P_ni, self.setups[a].dO_ii)
nblocks = self.get_optimal_number_of_blocks(self.blocking)
overlap = MatrixOperator(self.ksl, nblocks, self.async, True)
self.psit_nG = overlap.matrix_multiply(C_nn.T.copy(), self.psit_nG, self.P_ani)
D_nn = overlap.calculate_matrix_elements(self.psit_nG, \
self.P_ani, S, dS).T.copy() # transpose to get <psit_m|A|psit_n>
tri2full(D_nn, 'U') # upper to lower...
if self.bd.comm.rank == 0:
self.gd.comm.broadcast(D_nn, 0)
self.bd.comm.broadcast(D_nn, 0)
if memstats:
self.mem_test = record_memory()
# D_nn = C_nn^dag * S_nn * C_nn = W_n since Q_nn^dag = Q_nn^(-1)
D0_nn = np.dot(C_nn.T.conj(), np.dot(S_nn, C_nn))
self.assertAlmostEqual(np.abs(D0_nn-np.diag(W_n)).max(), 0, 9)
self.check_and_plot(D_nn, D0_nn, 9, 'trivial,diagonalize')
def test_multiply_nonhermitian(self):
alpha = np.random.normal(size=1).astype(self.dtype)
if self.dtype == complex:
alpha += 1j*np.random.normal(size=1)
world.broadcast(alpha, 0)
# Known starting point of S_nn = <psit_m|S|psit_n>
S_NN = alpha*self.S0_nn
if self.dtype == complex:
C_NN = np.random.uniform(size=self.nbands**2) * \
np.exp(1j*np.random.uniform(0,2*np.pi,size=self.nbands**2))
else:
C_NN = np.random.normal(size=self.nbands**2)
C_NN = C_NN.reshape((self.nbands,self.nbands)) / np.linalg.norm(C_NN,2)
world.broadcast(C_NN, 0)
# Set up Hermitian overlap operator:
S = lambda x: alpha*x
dS = lambda a, P_ni: np.dot(alpha*P_ni, self.setups[a].dO_ii)
nblocks = self.get_optimal_number_of_blocks(self.blocking)
overlap = MatrixOperator(self.ksl, nblocks, self.async, False)
if 0: #XXX non-hermitian case so Nn2nn not just uplo='L' but rather 'G'
blockcomm = self.ksl.nndescriptor.blacsgrid.comm
self.ksl.Nn2nn = Redistributor(blockcomm, self.ksl.Nndescriptor,
self.ksl.nndescriptor)
if self.bd.comm.rank == 0 and self.gd.comm.rank == 0:
assert C_NN.shape == (self.bd.nbands,) * 2
tmp_NN = C_NN.T.copy() # C -> Fortran indexing
else:
tmp_NN = self.ksl.nndescriptor.as_serial().empty(dtype=C_NN.dtype)
C_nn = self.ksl.nndescriptor.distribute_from_master(tmp_NN)
self.psit_nG = overlap.matrix_multiply(C_nn, self.psit_nG, self.P_ani)
D_nn = overlap.calculate_matrix_elements(self.psit_nG, self.P_ani, S, dS)
if memstats:
self.mem_test = record_memory()
D_NN = self.ksl.nndescriptor.collect_on_master(D_nn)
if self.bd.comm.rank == 0 and self.gd.comm.rank == 0:
assert D_NN.shape == (self.bd.nbands,) * 2
D_NN = D_NN.T.copy() # Fortran -> C indexing
else:
assert D_NN.nbytes == 0
D_NN = np.empty((self.bd.nbands,) * 2, dtype=D_NN.dtype)
if self.bd.comm.rank == 0:
self.gd.comm.broadcast(D_NN, 0)
self.bd.comm.broadcast(D_NN, 0)
# D_nn = C_nn^dag * S_nn * C_nn
D0_NN = np.dot(C_NN.T.conj(), np.dot(S_NN, C_NN))
self.check_and_plot(D_NN, D0_NN, 9, 'multiply,nonhermitian')
def test_multiply_randomized(self):
# Known starting point of S_nn = <psit_m|S|psit_n>
S_NN = self.S0_nn
if self.dtype == complex:
C_NN = np.random.uniform(size=self.nbands**2) * \
np.exp(1j*np.random.uniform(0,2*np.pi,size=self.nbands**2))
else:
C_NN = np.random.normal(size=self.nbands**2)
C_NN = C_NN.reshape((self.nbands,self.nbands)) / np.linalg.norm(C_NN,2)
world.broadcast(C_NN, 0)
# Set up Hermitian overlap operator:
S = lambda x: x
dS = lambda a, P_ni: np.dot(P_ni, self.setups[a].dO_ii)
nblocks = self.get_optimal_number_of_blocks(self.blocking)
overlap = MatrixOperator(self.ksl, nblocks, self.async, True)
if self.bd.comm.rank == 0 and self.gd.comm.rank == 0:
assert C_NN.shape == (self.bd.nbands,) * 2
tmp_NN = C_NN.T.copy() # C -> Fortran indexing
else:
tmp_NN = self.ksl.nndescriptor.as_serial().empty(dtype=C_NN.dtype)
C_nn = self.ksl.nndescriptor.distribute_from_master(tmp_NN)
self.psit_nG = overlap.matrix_multiply(C_nn, self.psit_nG, self.P_ani)
D_nn = overlap.calculate_matrix_elements(self.psit_nG, self.P_ani, S, dS)
if memstats:
self.mem_test = record_memory()
D_NN = self.ksl.nndescriptor.collect_on_master(D_nn)
if self.bd.comm.rank == 0 and self.gd.comm.rank == 0:
assert D_NN.shape == (self.bd.nbands,) * 2
D_NN = D_NN.T.copy() # Fortran -> C indexing
tri2full(D_NN, 'U') # upper to lower...
else:
assert D_NN.nbytes == 0
D_NN = np.empty((self.bd.nbands,) * 2, dtype=D_NN.dtype)
if self.bd.comm.rank == 0:
self.gd.comm.broadcast(D_NN, 0)
self.bd.comm.broadcast(D_NN, 0)
# D_nn = C_nn^dag * S_nn * C_nn
D0_NN = np.dot(C_NN.T.conj(), np.dot(S_NN, C_NN))
self.check_and_plot(D_NN, D0_NN, 9, 'multiply,randomized')
def run(psit_mG):
overlap = MatrixOperator(ksl, K)
if 0:
overlap.work1_xG = work1_xG
overlap.work2_xG = work2_xG
#S_nn = np.empty((N, N))
def S(x):
return x
dS_aii = {0: np.ones((2, 2)) * 0.123, 1: np.ones((3, 3)) * 0.321}
def dS(a, P_ni):
return np.dot(P_ni, dS_aii[a])
S_nn = overlap.calculate_matrix_elements(psit_mG, P_ani, S, dS)
t1 = time()
if world.rank == 0:
print(S_nn.round(5))
inverse_cholesky(S_nn)
C_nn = S_nn
t2 = time()
if world.rank == 0:
print('Cholesky Time %f' % (t2 - t1))
# Distribute matrix:
world.broadcast(C_nn, 0)
psit_mG = overlap.matrix_multiply(C_nn, psit_mG, P_ani)
if world.rank == 0:
print('Made it past matrix multiply')
# Check:
S_nn = overlap.calculate_matrix_elements(psit_mG, P_ani, S, dS)
assert not (P_ani[0] - psit_mG[:, :2, 0, 0]).round(10).any()
assert not (P_ani[1] - psit_mG[:, -1, -1, -3:]).round(10).any()
if world.rank == 0:
for n in range(N):
assert abs(S_nn[n, n] - 1.0) < 1e-10
assert not S_nn[n + 1:, n].round(10).any()
return psit_mG
def run(psit_mG):
overlap = MatrixOperator(ksl, J)
def H(psit_xG):
Htpsit_xG = np.empty_like(psit_xG)
kin(psit_xG, Htpsit_xG)
for psit_G, y_G in zip(psit_xG, Htpsit_xG):
y_G += vt_G * psit_G
return Htpsit_xG
dH_aii = {0: np.ones((2, 2)) * 0.123, 1: np.ones((3, 3)) * 0.321}
def dH(a, P_ni):
return np.dot(P_ni, dH_aii[a])
H_nn = overlap.calculate_matrix_elements(psit_mG, P_ani, H, dH)
t1 = time()
if world.rank == 0:
eps_n, H_nn = np.linalg.eigh(H_nn)
H_nn = np.ascontiguousarray(H_nn.T)
t2 = time()
if world.rank == 0:
print('Diagonalization Time %f' % (t2 - t1))
print(eps_n)
# Distribute matrix:
world.broadcast(H_nn, 0)
psit_mG = overlap.matrix_multiply(H_nn, psit_mG, P_ani)
if world.rank == 0:
print('Made it past matrix multiply')
# Check:
assert not (P_ani[0] - psit_mG[:, :2, 0, 0]).round(10).any()
assert not (P_ani[1] - psit_mG[:, -1, -1, -3:]).round(10).any()
H_nn = overlap.calculate_matrix_elements(psit_mG, P_ani, H, dH)
if world.rank == 0:
for n in range(N):
assert abs(H_nn[n, n] - eps_n[n]) < 2e-8
assert not H_nn[n + 1:, n].round(8).any()
return psit_mG
def test_trivial_diagonalize(self): #XXX XXX XXX
# Known starting point of S_nn = <psit_m|S|psit_n>
S_nn = self.S0_nn
# Eigenvector decomposition S_nn = V_nn * W_nn * V_nn^dag
# Utilize the fact that they are analytically known (cf. Maple)
W_n = np.zeros(self.nbands).astype(self.dtype)
W_n[1] = (1. + self.Qtotal) * self.nbands * (self.nbands - 1) / 2.
# Set up Hermitian overlap operator:
S = lambda x: x
dS = lambda a, P_ni: np.dot(P_ni, self.setups[a].dO_ii)
nblocks = self.get_optimal_number_of_blocks(self.blocking)
overlap = MatrixOperator(self.ksl, nblocks, self.async, True)
S_nn = overlap.calculate_matrix_elements(self.psit_nG, self.P_ani, S, dS)
eps_N = self.bd.empty(global_array=True) # XXX dtype?
C_nn = self.ksl.nndescriptor.empty(dtype=S_nn.dtype)
self.ksl.nndescriptor.diagonalize_dc(S_nn, C_nn, eps_N, 'L')
self.assertAlmostEqual(np.abs(np.sort(eps_N)-np.sort(W_n)).max(), 0, 9)
#eps_n = self.bd.empty()
#self.bd.distribute(eps_N, eps_n) # XXX only blocked groups, right?
# Rotate wavefunctions to diagonalize the overlap
self.psit_nG = overlap.matrix_multiply(C_nn, self.psit_nG, self.P_ani)
# Recaulculate the overlap matrix, which should now be diagonal
D_nn = overlap.calculate_matrix_elements(self.psit_nG, self.P_ani, S, dS)
D_NN = self.ksl.nndescriptor.collect_on_master(D_nn)
if self.bd.comm.rank == 0 and self.gd.comm.rank == 0:
assert D_NN.shape == (self.bd.nbands,) * 2
D_NN = D_NN.T.copy() # Fortran -> C indexing
tri2full(D_NN, 'U') # upper to lower...
else:
assert D_NN.nbytes == 0
D_NN = np.empty((self.bd.nbands,) * 2, dtype=D_NN.dtype)
if self.bd.comm.rank == 0:
self.gd.comm.broadcast(D_NN, 0)
self.bd.comm.broadcast(D_NN, 0)
D0_NN = np.diag(eps_N)
self.check_and_plot(D_NN, D0_NN, 9, 'trivial,diagonalize')
def run(psit_mG):
overlap = MatrixOperator(ksl, K)
if 0:
overlap.work1_xG = work1_xG
overlap.work2_xG = work2_xG
#S_nn = np.empty((N, N))
def S(x):
return x
dS_aii = {0: np.ones((2, 2)) * 0.123, 1: np.ones((3, 3)) * 0.321}
def dS(a, P_ni):
return np.dot(P_ni, dS_aii[a])
S_nn = overlap.calculate_matrix_elements(psit_mG, P_ani, S, dS)
t1 = time()
if world.rank == 0:
print S_nn.round(5)
inverse_cholesky(S_nn)
C_nn = S_nn
t2 = time()
if world.rank == 0:
print 'Cholesky Time %f' % (t2-t1)
# Distribute matrix:
world.broadcast(C_nn, 0)
psit_mG = overlap.matrix_multiply(C_nn, psit_mG, P_ani)
if world.rank == 0:
print 'Made it past matrix multiply'
# Check:
S_nn = overlap.calculate_matrix_elements(psit_mG, P_ani, S, dS)
assert not(P_ani[0] - psit_mG[:, :2, 0, 0]).round(10).any()
assert not(P_ani[1] - psit_mG[:, -1, -1, -3:]).round(10).any()
if world.rank == 0:
for n in range(N):
assert abs(S_nn[n, n] - 1.0) < 1e-10
assert not S_nn[n + 1:, n].round(10).any()
return psit_mG
def run(psit_mG):
overlap = MatrixOperator(ksl, J)
def H(psit_xG):
Htpsit_xG = np.empty_like(psit_xG)
kin(psit_xG, Htpsit_xG)
for psit_G, y_G in zip(psit_xG, Htpsit_xG):
y_G += vt_G * psit_G
return Htpsit_xG
dH_aii = {0: np.ones((2, 2)) * 0.123, 1: np.ones((3, 3)) * 0.321}
def dH(a, P_ni):
return np.dot(P_ni, dH_aii[a])
H_nn = overlap.calculate_matrix_elements(psit_mG, P_ani, H, dH)
t1 = time()
if world.rank == 0:
eps_n, H_nn = np.linalg.eigh(H_nn)
H_nn = np.ascontiguousarray(H_nn.T)
t2 = time()
if world.rank == 0:
print('Diagonalization Time %f' % (t2-t1))
print(eps_n)
# Distribute matrix:
world.broadcast(H_nn, 0)
psit_mG = overlap.matrix_multiply(H_nn, psit_mG, P_ani)
if world.rank == 0:
print('Made it past matrix multiply')
# Check:
assert not(P_ani[0] - psit_mG[:, :2, 0, 0]).round(10).any()
assert not(P_ani[1] - psit_mG[:, -1, -1, -3:]).round(10).any()
H_nn = overlap.calculate_matrix_elements(psit_mG, P_ani, H, dH)
if world.rank == 0:
for n in range(N):
assert abs(H_nn[n, n] - eps_n[n]) < 1.5e-8
assert not H_nn[n + 1:, n].round(8).any()
return psit_mG
def test_trivial_cholesky(self):
# Set up Hermitian overlap operator:
S = lambda x: x
dS = lambda a, P_ni: np.dot(P_ni, self.setups[a].dO_ii)
nblocks = self.get_optimal_number_of_blocks(self.blocking)
overlap = MatrixOperator(self.ksl, nblocks, self.async, True)
S_nn = overlap.calculate_matrix_elements(self.psit_nG, self.P_ani, S, dS)
# Known starting point of SI_nn = <psit_m|S+alpha*I|psit_n>
I_nn = self.ksl.nndescriptor.empty(dtype=S_nn.dtype)
scalapack_set(self.ksl.nndescriptor, I_nn, 0.0, 1.0, 'L')
alpha = 1e-3 # shift eigenvalues away from zero
C_nn = S_nn + alpha * I_nn
self.ksl.nndescriptor.inverse_cholesky(C_nn, 'L')
self.psit_nG = overlap.matrix_multiply(C_nn, self.psit_nG, self.P_ani)
D_nn = overlap.calculate_matrix_elements(self.psit_nG, self.P_ani, S, dS)
D_NN = self.ksl.nndescriptor.collect_on_master(D_nn)
if self.bd.comm.rank == 0 and self.gd.comm.rank == 0:
assert D_NN.shape == (self.bd.nbands,) * 2
D_NN = D_NN.T.copy() # Fortran -> C indexing
tri2full(D_NN, 'U') # upper to lower..
else:
assert D_NN.nbytes == 0
D_NN = np.empty((self.bd.nbands,) * 2, dtype=D_NN.dtype)
if self.bd.comm.rank == 0:
self.gd.comm.broadcast(D_NN, 0)
self.bd.comm.broadcast(D_NN, 0)
# D_NN = C_NN^dag * S_NN * C_NN = I_NN - alpha * C_NN^dag * C_NN
I_NN = np.eye(self.bd.nbands)
C0_NN = np.linalg.inv(np.linalg.cholesky(self.S0_nn + alpha*I_NN).T.conj())
D0_NN = I_NN - alpha * np.dot(C0_NN.T.conj(), C0_NN)
self.check_and_plot(D_NN, D0_NN, 6, 'trivial,cholesky') #XXX precision
