def matrix_exponential(G_nn, dlt):
"""Computes the matrix exponential of an antihermitian operator
U = exp(i*dlt*G)
G_nn: ndarray
anti-hermitian (skew-symmetric) matrix.
dlt: float
scaling factor for G_nn.
"""
ndim = G_nn.shape[1]
w_n = np.zeros((ndim), dtype=float)
V_nn = np.zeros((ndim, ndim), dtype=complex)
O_nn = np.zeros((ndim, ndim), dtype=complex)
if G_nn.dtype == complex:
V_nn = 1j * G_nn.real - G_nn.imag
else:
V_nn = 1j * G_nn.real
diagonalize(V_nn, w_n)
#
O_nn = np.diag(np.exp(1j * dlt * w_n))
#
if G_nn.dtype == complex:
U_nn = np.dot(V_nn.T.conj(), np.dot(O_nn, V_nn)).copy()
else:
U_nn = np.dot(V_nn.T.conj(), np.dot(O_nn, V_nn)).real.copy()
#
return U_nn
def diagonalize(self, istart=None, jend=None, energy_range=None):
"""Evaluate Eigenvectors and Eigenvalues:"""
map, kss = self.get_map(istart, jend, energy_range)
nij = len(kss)
if map is None:
ApB = self.ApB.copy()
AmB = self.AmB.copy()
nij = len(kss)
else:
ApB = np.empty((nij, nij))
AmB = np.empty((nij, nij))
for ij in range(nij):
for kq in range(nij):
ApB[ij,kq] = self.ApB[map[ij],map[kq]]
AmB[ij,kq] = self.AmB[map[ij],map[kq]]
# the occupation matrix
C = np.empty((nij,))
for ij in range(nij):
C[ij] = 1. / kss[ij].fij
S = C * inv(AmB) * C
S = sqrt_matrix(inv(S).copy())
# get Omega matrix
M = np.zeros(ApB.shape)
gemm(1.0, ApB, S, 0.0, M)
self.eigenvectors = np.zeros(ApB.shape)
gemm(1.0, S, M, 0.0, self.eigenvectors)
self.eigenvalues = np.zeros((len(kss)))
self.kss = kss
diagonalize(self.eigenvectors, self.eigenvalues)
def matrix_exponential(G_nn, dlt):
"""Computes the matrix exponential of an antihermitian operator
U = exp(i*dlt*G)
G_nn: ndarray
anti-hermitian (skew-symmetric) matrix.
dlt: float
scaling factor for G_nn.
"""
ndim = G_nn.shape[1]
w_n = np.zeros((ndim), dtype=float)
V_nn = np.zeros((ndim, ndim), dtype=complex)
O_nn = np.zeros((ndim, ndim), dtype=complex)
if G_nn.dtype==complex:
V_nn = 1j*G_nn.real - G_nn.imag
else:
V_nn = 1j*G_nn.real
diagonalize(V_nn, w_n)
#
O_nn = np.diag(np.exp(1j*dlt*w_n))
#
if G_nn.dtype==complex:
U_nn = np.dot(V_nn.T.conj(),np.dot(O_nn, V_nn)).copy()
else:
U_nn = np.dot(V_nn.T.conj(),np.dot(O_nn, V_nn)).real.copy()
#
return U_nn
def diagonalize(self, H_sS):
if self.coupling: # Non-Hermitian matrix can only use linalg.eig
self.printtxt('Use numpy.linalg.eig')
H_SS = np.zeros((self.nS, self.nS), dtype=complex)
if self.nS % world.size == 0:
world.all_gather(H_sS, H_SS)
else:
H_SS = gatherv(H_sS)
self.w_S, self.v_SS = np.linalg.eig(H_SS)
self.par_save('v_SS', 'v_SS', self.v_SS[self.nS_start:self.nS_end, :].copy())
else:
if world.size == 1:
self.printtxt('Use lapack.')
from gpaw.utilities.lapack import diagonalize
self.w_S = np.zeros(self.nS)
H_SS = H_sS
diagonalize(H_SS, self.w_S)
self.v_SS = H_SS.conj() # eigenvectors in the rows, transposed later
else:
self.printtxt('Use scalapack')
self.w_S, self.v_sS = self.scalapack_diagonalize(H_sS)
self.v_SS = self.v_sS # just use the same name
self.par_save('v_SS', 'v_SS', self.v_SS)
return
def ortho(W_nn, maxerr=1E-10):
""" Orthogonalizes the column vectors of a matrix by Symmetric
Loewdin orthogonalization
W_nn: ndarray
unorthonormal matrix.
maxerr: float
maximum error for using explicit diagonalization.
"""
ndim = np.shape(W_nn)[1]
# overlap matrix
O_nn = np.dot(W_nn, W_nn.T.conj())
# check error in orthonormality
err = np.sum(np.abs(O_nn - np.eye(ndim)))
if (err < maxerr):
# perturbative Symmetric-Loewdin
X_nn = 1.5 * np.eye(ndim) - 0.5 * O_nn
else:
# diagonalization
n_n = np.zeros(ndim, dtype=float)
diagonalize(O_nn, n_n)
U_nn = O_nn.T.conj().copy()
nsqrt_n = np.diag(1.0 / np.sqrt(n_n))
X_nn = np.dot(np.dot(U_nn, nsqrt_n), U_nn.T.conj())
# apply orthonormalizing transformation
O_nn = np.dot(X_nn, W_nn)
return O_nn
def diagonalize(self, H_sS):
if self.coupling: # Non-Hermitian matrix can only use linalg.eig
self.printtxt('Use numpy.linalg.eig')
H_SS = np.zeros((self.nS, self.nS), dtype=complex)
if self.nS % world.size == 0:
world.all_gather(H_sS, H_SS)
else:
H_SS = gatherv(H_sS)
self.w_S, self.v_SS = np.linalg.eig(H_SS)
self.par_save('v_SS', 'v_SS',
self.v_SS[self.nS_start:self.nS_end, :].copy())
else:
if world.size == 1:
self.printtxt('Use lapack.')
from gpaw.utilities.lapack import diagonalize
self.w_S = np.zeros(self.nS)
H_SS = H_sS
diagonalize(H_SS, self.w_S)
self.v_SS = H_SS.conj(
) # eigenvectors in the rows, transposed later
else:
self.printtxt('Use scalapack')
self.w_S, self.v_sS = self.scalapack_diagonalize(H_sS)
self.v_SS = self.v_sS # just use the same name
self.par_save('v_SS', 'v_SS', self.v_SS)
return
def diagonalize(self,
istart=None,
jend=None,
energy_range=None,
TDA=False):
"""Evaluate Eigenvectors and Eigenvalues"""
ApB, AmB = self.map(istart, jend, energy_range)
nij = len(self.kss)
if TDA:
# Tamm-Dancoff approximation (B=0)
self.eigenvectors = 0.5 * (ApB + AmB)
eigenvalues = np.zeros((nij))
diagonalize(self.eigenvectors, eigenvalues)
self.eigenvalues = eigenvalues**2
else:
# the occupation matrix
C = np.empty((nij, ))
for ij in range(nij):
C[ij] = 1. / self.kss[ij].fij
S = C * inv(AmB) * C
S = sqrt_matrix(inv(S).copy())
# get Omega matrix
M = np.zeros(ApB.shape)
gemm(1.0, ApB, S, 0.0, M)
self.eigenvectors = np.zeros(ApB.shape)
gemm(1.0, S, M, 0.0, self.eigenvectors)
self.eigenvalues = np.zeros((nij))
diagonalize(self.eigenvectors, self.eigenvalues)
def diagonalize(self, istart=None, jend=None, energy_range=None, TDA=False):
"""Evaluate Eigenvectors and Eigenvalues"""
ApB, AmB = self.map(istart, jend, energy_range)
nij = len(self.kss)
if TDA:
# Tamm-Dancoff approximation (B=0)
self.eigenvectors = 0.5 * (ApB + AmB)
eigenvalues = np.zeros((nij))
diagonalize(self.eigenvectors, eigenvalues)
self.eigenvalues = eigenvalues ** 2
else:
# the occupation matrix
C = np.empty((nij,))
for ij in range(nij):
C[ij] = 1.0 / self.kss[ij].fij
S = C * inv(AmB) * C
S = sqrt_matrix(inv(S).copy())
# get Omega matrix
M = np.zeros(ApB.shape)
gemm(1.0, ApB, S, 0.0, M)
self.eigenvectors = np.zeros(ApB.shape)
gemm(1.0, S, M, 0.0, self.eigenvectors)
self.eigenvalues = np.zeros((nij))
diagonalize(self.eigenvectors, self.eigenvalues)
def diagonalize(self):
print('Diagonalizing Hamiltonian', file=self.fd)
"""The t and T represent local and global
eigenstates indices respectively
"""
# Non-Hermitian matrix can only use linalg.eig
if not self.td:
print(' Using numpy.linalg.eig...', file=self.fd)
print(' Eliminated %s pair orbitals' % len(self.excludef_S),
file=self.fd)
self.H_SS = self.collect_A_SS(self.H_sS)
self.w_T = np.zeros(self.nS - len(self.excludef_S), complex)
if world.rank == 0:
self.H_SS = np.delete(self.H_SS, self.excludef_S, axis=0)
self.H_SS = np.delete(self.H_SS, self.excludef_S, axis=1)
self.w_T, self.v_ST = np.linalg.eig(self.H_SS)
world.broadcast(self.w_T, 0)
self.df_S = np.delete(self.df_S, self.excludef_S)
self.rhoG0_S = np.delete(self.rhoG0_S, self.excludef_S)
# Here the eigenvectors are returned as complex conjugated rows
else:
if world.size == 1:
print(' Using lapack...', file=self.fd)
from gpaw.utilities.lapack import diagonalize
self.w_T = np.zeros(self.nS)
diagonalize(self.H_sS, self.w_T)
self.v_St = self.H_sS.conj().T
else:
print(' Using scalapack...', file=self.fd)
nS = self.nS
ns = -(-self.kd.nbzkpts // world.size) * (self.nv * self.nc *
self.spins *
(self.spinors + 1)**2)
grid = BlacsGrid(world, world.size, 1)
desc = grid.new_descriptor(nS, nS, ns, nS)
desc2 = grid.new_descriptor(nS, nS, 2, 2)
H_tmp = desc2.zeros(dtype=complex)
r = Redistributor(world, desc, desc2)
r.redistribute(self.H_sS, H_tmp)
self.w_T = np.empty(nS)
v_tmp = desc2.empty(dtype=complex)
desc2.diagonalize_dc(H_tmp, v_tmp, self.w_T)
r = Redistributor(grid.comm, desc2, desc)
self.v_St = desc.zeros(dtype=complex)
r.redistribute(v_tmp, self.v_St)
self.v_St = self.v_St.conj().T
if self.write_v and self.td:
# Cannot use par_save without td
self.par_save('v_TS.ulm', 'v_TS', self.v_St.T)
return
def main(nbands=1000, mprocs=2, mb=64):
# Set-up BlacsGrud
grid = BlacsGrid(world, mprocs, mprocs)
# Create descriptor
nndesc = grid.new_descriptor(nbands, nbands, mb, mb)
H_nn = nndesc.empty(
dtype=float) # outside the BlacsGrid these are size zero
C_nn = nndesc.empty(
dtype=float) # outside the BlacsGrid these are size zero
eps_N = np.empty((nbands),
dtype=float) # replicated array on all MPI tasks
# Fill ScaLAPACK array
alpha = 0.1 # off-diagonal
beta = 75.0 # diagonal
uplo = 'L' # lower-triangular
scalapack_set(nndesc, H_nn, alpha, beta, uplo)
scalapack_zero(nndesc, H_nn, switch_uplo[uplo])
t1 = time()
# either interface will work, we recommend use the latter interface
# scalapack_diagonalize_dc(nndesc, H_nn.copy(), C_nn, eps_N, 'L')
nndesc.diagonalize_dc(H_nn.copy(), C_nn, eps_N)
t2 = time()
world.broadcast(eps_N, 0) # all MPI tasks now have eps_N
world.barrier() # wait for everyone to finish
if rank == 0:
print('ScaLAPACK diagonalize_dc', t2 - t1)
# Create replicated NumPy array
diagonal = np.eye(nbands, dtype=float)
offdiagonal = np.tril(np.ones((nbands, nbands)), -1)
H0 = beta * diagonal + alpha * offdiagonal
E0 = np.empty((nbands), dtype=float)
t1 = time()
diagonalize(H0, E0)
t2 = time()
if rank == 0:
print('LAPACK diagonalize', t2 - t1)
delta = abs(E0 - eps_N).max()
if rank == 0:
print(delta)
assert delta < tol
def diagonalize(self, istart=None, jend=None, energy_range=None):
"""Evaluate Eigenvectors and Eigenvalues:"""
map, kss = self.get_map(istart, jend, energy_range)
nij = len(kss)
if map is None:
evec = self.full.copy()
else:
evec = np.zeros((nij,nij))
for ij in range(nij):
for kq in range(nij):
evec[ij,kq] = self.full[map[ij],map[kq]]
self.eigenvectors = evec
self.eigenvalues = np.zeros((len(kss)))
self.kss = kss
diagonalize(self.eigenvectors, self.eigenvalues)
def diagonalize(self, istart=None, jend=None, energy_range=None):
"""Evaluate Eigenvectors and Eigenvalues:"""
map, kss = self.get_map(istart, jend, energy_range)
nij = len(kss)
if map is None:
evec = self.full.copy()
else:
evec = np.zeros((nij, nij))
for ij in range(nij):
for kq in range(nij):
evec[ij, kq] = self.full[map[ij], map[kq]]
self.eigenvectors = evec
self.eigenvalues = np.zeros((len(kss)))
self.kss = kss
diagonalize(self.eigenvectors, self.eigenvalues)
def main(nbands=1000, mprocs=2, mb=64):
# Set-up BlacsGrud
grid = BlacsGrid(world, mprocs, mprocs)
# Create descriptor
nndesc = grid.new_descriptor(nbands, nbands, mb, mb)
H_nn = nndesc.empty(dtype=float) # outside the BlacsGrid these are size zero
C_nn = nndesc.empty(dtype=float) # outside the BlacsGrid these are size zero
eps_N = np.empty((nbands), dtype=float) # replicated array on all MPI tasks
# Fill ScaLAPACK array
alpha = 0.1 # off-diagonal
beta = 75.0 # diagonal
uplo = 'L' # lower-triangular
scalapack_set(nndesc, H_nn, alpha, beta, uplo)
scalapack_zero(nndesc, H_nn, switch_uplo[uplo])
t1 = time()
# either interface will work, we recommend use the latter interface
# scalapack_diagonalize_dc(nndesc, H_nn.copy(), C_nn, eps_N, 'L')
nndesc.diagonalize_dc(H_nn.copy(), C_nn, eps_N)
t2 = time()
world.broadcast(eps_N, 0) # all MPI tasks now have eps_N
world.barrier() # wait for everyone to finish
if rank == 0:
print('ScaLAPACK diagonalize_dc', t2-t1)
# Create replicated NumPy array
diagonal = np.eye(nbands,dtype=float)
offdiagonal = np.tril(np.ones((nbands,nbands)), -1)
H0 = beta*diagonal + alpha*offdiagonal
E0 = np.empty((nbands), dtype=float)
t1 = time()
diagonalize(H0,E0)
t2 = time()
if rank == 0:
print('LAPACK diagonalize', t2-t1)
delta = abs(E0-eps_N).max()
if rank == 0:
print(delta)
assert delta < tol
def diagonalize(self, istart=None, jend=None, energy_range=None,
TDA=False):
"""Evaluate Eigenvectors and Eigenvalues:"""
if TDA:
raise NotImplementedError
map, kss = self.get_map(istart, jend, energy_range)
nij = len(kss)
if map is None:
evec = self.full.copy()
else:
evec = np.zeros((nij,nij))
for ij in range(nij):
for kq in range(nij):
evec[ij,kq] = self.full[map[ij],map[kq]]
assert(len(evec) > 0)
self.eigenvectors = evec
self.eigenvalues = np.zeros((len(kss)))
self.kss = kss
diagonalize(self.eigenvectors, self.eigenvalues)
def main(i,seed=42,dtype=float):
if (dtype==complex):
epsilon = 0.1j
else:
epsilon = 0.1
x = i + 1
N = x*100
print "N =",N
H0 = np.zeros((N,N),dtype=dtype) + gen.rand(*(N,N))
H1 = H0 + epsilon*np.tri(N,N, k=-1)
W0 = np.zeros((N))
Z0 = np.zeros_like(H0)
t0 = time()
diagonalize(H1, W0)
t1 = time() - t0
print "diagonalize", t1
t2 = time()
diagonalize_mr3(H1, W0, Z0)
t3 = time() - t2
print "diagonalize_mr3",t3
# diagonalize_mr3 must be faster than diagonalize
assert(t3 < t1)
def main(i, seed=42, dtype=float):
if (dtype == complex):
epsilon = 0.1j
else:
epsilon = 0.1
x = i + 1
N = x * 100
print "N =", N
H0 = np.zeros((N, N), dtype=dtype) + gen.rand(*(N, N))
H1 = H0 + epsilon * np.tri(N, N, k=-1)
W0 = np.zeros((N))
Z0 = np.zeros_like(H0)
t0 = time()
diagonalize(H1, W0)
t1 = time() - t0
print "diagonalize", t1
t2 = time()
diagonalize_mr3(H1, W0, Z0)
t3 = time() - t2
print "diagonalize_mr3", t3
# diagonalize_mr3 must be faster than diagonalize
assert (t3 < t1)
def diagonalize(self, istart=None, jend=None, energy_range=None,
TDA=False):
"""Evaluate Eigenvectors and Eigenvalues:"""
if TDA:
raise NotImplementedError
map, kss = self.get_map(istart, jend, energy_range)
nij = len(kss)
if map is None:
evec = self.full.copy()
else:
evec = np.zeros((nij, nij))
for ij in range(nij):
for kq in range(nij):
evec[ij, kq] = self.full[map[ij], map[kq]]
assert(len(evec) > 0)
self.eigenvectors = evec
self.eigenvalues = np.zeros((len(kss)))
self.kss = kss
diagonalize(self.eigenvectors, self.eigenvalues)
def ortho(W_nn, maxerr=1E-10):
""" Orthogonalizes the column vectors of a matrix by Symmetric
Loewdin orthogonalization
W_nn: ndarray
unorthonormal matrix.
maxerr: float
maximum error for using explicit diagonalization.
"""
ndim = np.shape(W_nn)[1]
#
# overlap matrix
O_nn = np.dot(W_nn, W_nn.T.conj())
#
# check error in orthonormality
err = np.sum(np.abs(O_nn - np.eye(ndim)))
if (err < maxerr):
#
# perturbative Symmetric-Loewdin
X_nn= 1.5*np.eye(ndim) - 0.5*O_nn
else:
#
# diagonalization
n_n = np.zeros(ndim, dtype=float)
diagonalize(O_nn, n_n)
U_nn = O_nn.T.conj().copy()
nsqrt_n = np.diag(1.0/np.sqrt(n_n))
X_nn = np.dot(np.dot(U_nn, nsqrt_n), U_nn.T.conj())
#
# apply orthonormalizing transformation
O_nn = np.dot(X_nn, W_nn)
return O_nn
return eps, B
elif nodes == 'all':
if rank != 0:
B = np.zeros((N, N))
world.broadcast(B, 0)
return eps, B
# generate a matrix
N = 512
A = np.arange(N**2, dtype=float).reshape(N, N)
for i in range(N):
for j in range(i, N):
A[i, j] = A[j, i]
# diagonalize
eps, B = scal_diagonalize(A)
check = 1
if check and rank == 0:
# check whether it gives the same result with lapack
eps1 = np.zeros(N)
diagonalize(A, eps1)
assert np.abs(eps - eps1).sum() < 1e-6
for i in range(N // size):
# the eigenvectors are row of the matrix, it can be differ by a minus sign.
if np.abs(A[i, :] - B[i, :]).sum() > 1e-6:
if np.abs(A[i, :] + B[i, :]).sum() > 1e-6:
raise ValueError('Check !')
def main(N=72, seed=42, mprocs=2, nprocs=2, dtype=float):
gen = np.random.RandomState(seed)
grid = BlacsGrid(world, mprocs, nprocs)
if (dtype == complex):
epsilon = 1.0j
else:
epsilon = 0.0
# Create descriptors for matrices on master:
glob = grid.new_descriptor(N, N, N, N)
# print globA.asarray()
# Populate matrices local to master:
H0 = glob.zeros(dtype=dtype) + gen.rand(*glob.shape)
S0 = glob.zeros(dtype=dtype) + gen.rand(*glob.shape)
C0 = glob.empty(dtype=dtype)
if rank == 0:
# Complex case must have real numbers on the diagonal.
# We make a simple complex Hermitian matrix below.
H0 = H0 + epsilon * (0.1 * np.tri(N, N, k=-N // nprocs) +
0.3 * np.tri(N, N, k=-1))
S0 = S0 + epsilon * (0.2 * np.tri(N, N, k=-N // nprocs) +
0.4 * np.tri(N, N, k=-1))
# Make matrices symmetric
rk(1.0, H0.copy(), 0.0, H0)
rk(1.0, S0.copy(), 0.0, S0)
# Overlap matrix must be semi-positive definite
S0 = S0 + 50.0 * np.eye(N, N, 0)
# Hamiltonian is usually diagonally dominant
H0 = H0 + 75.0 * np.eye(N, N, 0)
C0 = S0.copy()
S0_inv = S0.copy()
# Local result matrices
W0 = np.empty((N), dtype=float)
W0_g = np.empty((N), dtype=float)
# Calculate eigenvalues / other serial results
if rank == 0:
diagonalize(H0.copy(), W0)
general_diagonalize(H0.copy(), W0_g, S0.copy())
inverse_cholesky(C0) # result returned in lower triangle
tri2full(S0_inv, 'L')
S0_inv = inv(S0_inv)
# tri2full(C0) # symmetrize
assert glob.check(H0) and glob.check(S0) and glob.check(C0)
# Create distributed destriptors with various block sizes:
dist = grid.new_descriptor(N, N, 8, 8)
# Distributed matrices:
# We can use empty here, but end up with garbage on
# on the other half of the triangle when we redistribute.
# This is fine because ScaLAPACK does not care.
H = dist.empty(dtype=dtype)
S = dist.empty(dtype=dtype)
Sinv = dist.empty(dtype=dtype)
Z = dist.empty(dtype=dtype)
C = dist.empty(dtype=dtype)
Sinv = dist.empty(dtype=dtype)
# Eigenvalues are non-BLACS matrices
W = np.empty((N), dtype=float)
W_dc = np.empty((N), dtype=float)
W_mr3 = np.empty((N), dtype=float)
W_g = np.empty((N), dtype=float)
W_g_dc = np.empty((N), dtype=float)
W_g_mr3 = np.empty((N), dtype=float)
Glob2dist = Redistributor(world, glob, dist)
Glob2dist.redistribute(H0, H, uplo='L')
Glob2dist.redistribute(S0, S, uplo='L')
Glob2dist.redistribute(S0, C, uplo='L') # C0 was previously overwritten
Glob2dist.redistribute(S0, Sinv, uplo='L')
# we don't test the expert drivers anymore since there
# might be a buffer overflow error
## scalapack_diagonalize_ex(dist, H.copy(), Z, W, 'L')
scalapack_diagonalize_dc(dist, H.copy(), Z, W_dc, 'L')
## scalapack_diagonalize_mr3(dist, H.copy(), Z, W_mr3, 'L')
## scalapack_general_diagonalize_ex(dist, H.copy(), S.copy(), Z, W_g, 'L')
scalapack_general_diagonalize_dc(dist, H.copy(), S.copy(), Z, W_g_dc, 'L')
## scalapack_general_diagonalize_mr3(dist, H.copy(), S.copy(), Z, W_g_mr3, 'L')
scalapack_inverse_cholesky(dist, C, 'L')
if dtype == complex: # Only supported for complex for now
scalapack_inverse(dist, Sinv, 'L')
# Undo redistribute
C_test = glob.empty(dtype=dtype)
Sinv_test = glob.empty(dtype=dtype)
Dist2glob = Redistributor(world, dist, glob)
Dist2glob.redistribute(C, C_test)
Dist2glob.redistribute(Sinv, Sinv_test)
if rank == 0:
## diag_ex_err = abs(W - W0).max()
diag_dc_err = abs(W_dc - W0).max()
## diag_mr3_err = abs(W_mr3 - W0).max()
## general_diag_ex_err = abs(W_g - W0_g).max()
general_diag_dc_err = abs(W_g_dc - W0_g).max()
## general_diag_mr3_err = abs(W_g_mr3 - W0_g).max()
inverse_chol_err = abs(C_test - C0).max()
tri2full(Sinv_test, 'L')
inverse_err = abs(Sinv_test - S0_inv).max()
## print 'diagonalize ex err', diag_ex_err
print('diagonalize dc err', diag_dc_err)
## print 'diagonalize mr3 err', diag_mr3_err
## print 'general diagonalize ex err', general_diag_ex_err
print('general diagonalize dc err', general_diag_dc_err)
## print 'general diagonalize mr3 err', general_diag_mr3_err
print('inverse chol err', inverse_chol_err)
if dtype == complex:
print('inverse err', inverse_err)
else:
## diag_ex_err = 0.0
diag_dc_err = 0.0
## diag_mr3_err = 0.0
## general_diag_ex_err = 0.0
general_diag_dc_err = 0.0
## general_diag_mr3_err = 0.0
inverse_chol_err = 0.0
inverse_err = 0.0
# We don't like exceptions on only one cpu
## diag_ex_err = world.sum(diag_ex_err)
diag_dc_err = world.sum(diag_dc_err)
## diag_mr3_err = world.sum(diag_mr3_err)
## general_diag_ex_err = world.sum(general_diag_ex_err)
general_diag_dc_err = world.sum(general_diag_dc_err)
## general_diag_mr3_err = world.sum(general_diag_mr3_err)
inverse_chol_err = world.sum(inverse_chol_err)
inverse_err = world.sum(inverse_err)
## assert diag_ex_err < tol
assert diag_dc_err < tol
## assert diag_mr3_err < tol
## assert general_diag_ex_err < tol
assert general_diag_dc_err < tol
## assert general_diag_mr3_err < tol
assert inverse_chol_err < tol
if dtype == complex:
assert inverse_err < tol
def _diagonalize(self, H_NN, eps_N):
"""Serial diagonalize via LAPACK."""
# This is replicated computation but ultimately avoids
# additional communication.
diagonalize(H_NN, eps_N)
def main(N=73, seed=42, mprocs=2, nprocs=2, dtype=float):
gen = np.random.RandomState(seed)
grid = BlacsGrid(world, mprocs, nprocs)
if (dtype==complex):
epsilon = 1.0j
else:
epsilon = 0.0
# Create descriptors for matrices on master:
glob = grid.new_descriptor(N, N, N, N)
# print globA.asarray()
# Populate matrices local to master:
H0 = glob.zeros(dtype=dtype) + gen.rand(*glob.shape)
S0 = glob.zeros(dtype=dtype) + gen.rand(*glob.shape)
C0 = glob.empty(dtype=dtype)
if rank == 0:
# Complex case must have real numbers on the diagonal.
# We make a simple complex Hermitian matrix below.
H0 = H0 + epsilon * (0.1*np.tri(N, N, k= -N // nprocs) + 0.3*np.tri(N, N, k=-1))
S0 = S0 + epsilon * (0.2*np.tri(N, N, k= -N // nprocs) + 0.4*np.tri(N, N, k=-1))
# Make matrices symmetric
rk(1.0, H0.copy(), 0.0, H0)
rk(1.0, S0.copy(), 0.0, S0)
# Overlap matrix must be semi-positive definite
S0 = S0 + 50.0*np.eye(N, N, 0)
# Hamiltonian is usually diagonally dominant
H0 = H0 + 75.0*np.eye(N, N, 0)
C0 = S0.copy()
# Local result matrices
W0 = np.empty((N),dtype=float)
W0_g = np.empty((N),dtype=float)
# Calculate eigenvalues
if rank == 0:
diagonalize(H0.copy(), W0)
general_diagonalize(H0.copy(), W0_g, S0.copy())
inverse_cholesky(C0) # result returned in lower triangle
# tri2full(C0) # symmetrize
assert glob.check(H0) and glob.check(S0) and glob.check(C0)
# Create distributed destriptors with various block sizes:
dist = grid.new_descriptor(N, N, 8, 8)
# Distributed matrices:
# We can use empty here, but end up with garbage on
# on the other half of the triangle when we redistribute.
# This is fine because ScaLAPACK does not care.
H = dist.empty(dtype=dtype)
S = dist.empty(dtype=dtype)
Z = dist.empty(dtype=dtype)
C = dist.empty(dtype=dtype)
# Eigenvalues are non-BLACS matrices
W = np.empty((N), dtype=float)
W_dc = np.empty((N), dtype=float)
W_mr3 = np.empty((N), dtype=float)
W_g = np.empty((N), dtype=float)
W_g_dc = np.empty((N), dtype=float)
W_g_mr3 = np.empty((N), dtype=float)
Glob2dist = Redistributor(world, glob, dist)
Glob2dist.redistribute(H0, H, uplo='L')
Glob2dist.redistribute(S0, S, uplo='L')
Glob2dist.redistribute(S0, C, uplo='L') # C0 was previously overwritten
# we don't test the expert drivers anymore since there
# might be a buffer overflow error
## scalapack_diagonalize_ex(dist, H.copy(), Z, W, 'L')
scalapack_diagonalize_dc(dist, H.copy(), Z, W_dc, 'L')
## scalapack_diagonalize_mr3(dist, H.copy(), Z, W_mr3, 'L')
## scalapack_general_diagonalize_ex(dist, H.copy(), S.copy(), Z, W_g, 'L')
scalapack_general_diagonalize_dc(dist, H.copy(), S.copy(), Z, W_g_dc, 'L')
## scalapack_general_diagonalize_mr3(dist, H.copy(), S.copy(), Z, W_g_mr3, 'L')
scalapack_inverse_cholesky(dist, C, 'L')
# Undo redistribute
C_test = glob.empty(dtype=dtype)
Dist2glob = Redistributor(world, dist, glob)
Dist2glob.redistribute(C, C_test)
if rank == 0:
## diag_ex_err = abs(W - W0).max()
diag_dc_err = abs(W_dc - W0).max()
## diag_mr3_err = abs(W_mr3 - W0).max()
## general_diag_ex_err = abs(W_g - W0_g).max()
general_diag_dc_err = abs(W_g_dc - W0_g).max()
## general_diag_mr3_err = abs(W_g_mr3 - W0_g).max()
inverse_chol_err = abs(C_test-C0).max()
## print 'diagonalize ex err', diag_ex_err
print 'diagonalize dc err', diag_dc_err
## print 'diagonalize mr3 err', diag_mr3_err
## print 'general diagonalize ex err', general_diag_ex_err
print 'general diagonalize dc err', general_diag_dc_err
## print 'general diagonalize mr3 err', general_diag_mr3_err
print 'inverse chol err', inverse_chol_err
else:
## diag_ex_err = 0.0
diag_dc_err = 0.0
## diag_mr3_err = 0.0
## general_diag_ex_err = 0.0
general_diag_dc_err = 0.0
## general_diag_mr3_err = 0.0
inverse_chol_err = 0.0
# We don't like exceptions on only one cpu
## diag_ex_err = world.sum(diag_ex_err)
diag_dc_err = world.sum(diag_dc_err)
## diag_mr3_err = world.sum(diag_mr3_err)
## general_diag_ex_err = world.sum(general_diag_ex_err)
general_diag_dc_err = world.sum(general_diag_dc_err)
## general_diag_mr3_err = world.sum(general_diag_mr3_err)
inverse_chol_err = world.sum(inverse_chol_err)
## assert diag_ex_err < tol
assert diag_dc_err < tol
## assert diag_mr3_err < tol
## assert general_diag_ex_err < tol
assert general_diag_dc_err < tol
## assert general_diag_mr3_err < tol
assert inverse_chol_err < tol
from __future__ import print_function import numpy as np tol = 1e-8 a = np.eye(3, dtype=complex) a[1:, 0] = 0.01j w0 = [0.98585786, 1.0, 1.01414214] # NumPy's Diagonalize from numpy.linalg import eigh w = eigh(a)[0] print(w) assert abs(w - w0).max() < tol # LAPACK's QR Diagonalize from gpaw.utilities.lapack import diagonalize diagonalize(a.copy(), w) print(w) assert abs(w - w0).max() < tol # LAPACK's MR3 Diagonalize # Requires Netlib LAPACK 3.2.1 or later # from gpaw.utilities.lapack import diagonalize_mr3 # z = np.zeros_like(a) # diagonalize_mr3(a, w, z) # print w # assert abs(w - w0).max() < tol
if rank != 0:
B = np.zeros((N, N))
world.broadcast(B, 0)
return eps, B
# generate a matrix
N = 512
A = np.arange(N**2,dtype=float).reshape(N,N)
for i in range(N):
for j in range(i,N):
A[i,j] = A[j,i]
# diagonalize
eps, B = scal_diagonalize(A)
check = 1
if check and rank == 0:
# check whether it gives the same result with lapack
eps1 = np.zeros(N)
diagonalize(A, eps1)
assert np.abs(eps-eps1).sum() < 1e-6
for i in range(N//size):
# the eigenvectors are row of the matrix, it can be differ by a minus sign.
if np.abs(A[i,:] - B[i,:]).sum() > 1e-6:
if np.abs(A[i,:] + B[i,:]).sum() > 1e-6:
raise ValueError('Check !')
def orthonormalize(self, wfs, kpt, psit_nG=None):
"""Orthonormalizes the vectors a_nG with respect to the overlap.
First, a Cholesky factorization C is done for the overlap
matrix S_nn = <a_nG | S | a_nG> = C*_nn C_nn Cholesky matrix C
is inverted and orthonormal vectors a_nG' are obtained as::
psit_nG' = inv(C_nn) psit_nG
__
~ _ \ -1 ~ _
psi (r) = ) C psi (r)
n /__ nm m
m
Parameters
----------
psit_nG: ndarray, input/output
On input the set of vectors to orthonormalize,
on output the overlap-orthonormalized vectors.
kpt: KPoint object:
k-point object from kpoint.py.
work_nG: ndarray
Optional work array for overlap matrix times psit_nG.
work_nn: ndarray
Optional work array for overlap matrix.
"""
self.timer.start('Orthonormalize')
if psit_nG is None:
psit_nG = kpt.psit_nG
P_ani = kpt.P_ani
self.timer.start('projections')
wfs.pt.integrate(psit_nG, P_ani, kpt.q)
self.timer.stop('projections')
# Construct the overlap matrix:
operator = wfs.matrixoperator
def S(psit_G):
return psit_G
def dS(a, P_ni):
return np.dot(P_ni, wfs.setups[a].dO_ii)
self.timer.start('calc_s_matrix')
S_nn = operator.calculate_matrix_elements(psit_nG, P_ani, S, dS)
self.timer.stop('calc_s_matrix')
orthonormalization_string = repr(self.ksl)
self.timer.start(orthonormalization_string)
#
if extra_parameters.get('sic', False):
#
# symmetric Loewdin Orthonormalization
tri2full(S_nn, UL='L', map=np.conj)
nrm_n = np.empty(S_nn.shape[0])
diagonalize(S_nn, nrm_n)
nrm_nn = np.diag(1.0/np.sqrt(nrm_n))
S_nn = np.dot(np.dot(S_nn.T.conj(), nrm_nn), S_nn)
else:
#
self.ksl.inverse_cholesky(S_nn)
# S_nn now contains the inverse of the Cholesky factorization.
# Let's call it something different:
C_nn = S_nn
del S_nn
self.timer.stop(orthonormalization_string)
self.timer.start('rotate_psi')
operator.matrix_multiply(C_nn, psit_nG, P_ani, out_nG=kpt.psit_nG)
self.timer.stop('rotate_psi')
self.timer.stop('Orthonormalize')
def orthonormalize(self, wfs, kpt, psit_nG=None):
"""Orthonormalizes the vectors a_nG with respect to the overlap.
First, a Cholesky factorization C is done for the overlap
matrix S_nn = <a_nG | S | a_nG> = C*_nn C_nn Cholesky matrix C
is inverted and orthonormal vectors a_nG' are obtained as::
psit_nG' = inv(C_nn) psit_nG
__
~ _ \ -1 ~ _
psi (r) = ) C psi (r)
n /__ nm m
m
Parameters
----------
psit_nG: ndarray, input/output
On input the set of vectors to orthonormalize,
on output the overlap-orthonormalized vectors.
kpt: KPoint object:
k-point object from kpoint.py.
work_nG: ndarray
Optional work array for overlap matrix times psit_nG.
work_nn: ndarray
Optional work array for overlap matrix.
"""
self.timer.start('Orthonormalize')
if psit_nG is None:
psit_nG = kpt.psit_nG
P_ani = kpt.P_ani
self.timer.start('projections')
wfs.pt.integrate(psit_nG, P_ani, kpt.q)
self.timer.stop('projections')
# Construct the overlap matrix:
operator = wfs.matrixoperator
def S(psit_G):
return psit_G
def dS(a, P_ni):
return np.dot(P_ni, wfs.setups[a].dO_ii)
self.timer.start('calc_s_matrix')
S_nn = operator.calculate_matrix_elements(psit_nG, P_ani, S, dS)
self.timer.stop('calc_s_matrix')
orthonormalization_string = repr(self.ksl)
self.timer.start(orthonormalization_string)
#
if extra_parameters.get('sic', False):
#
# symmetric Loewdin Orthonormalization
tri2full(S_nn, UL='L', map=np.conj)
nrm_n = np.empty(S_nn.shape[0])
diagonalize(S_nn, nrm_n)
nrm_nn = np.diag(1.0/np.sqrt(nrm_n))
S_nn = np.dot(np.dot(S_nn.T.conj(), nrm_nn), S_nn)
else:
#
self.ksl.inverse_cholesky(S_nn)
# S_nn now contains the inverse of the Cholesky factorization.
# Let's call it something different:
C_nn = S_nn
del S_nn
self.timer.stop(orthonormalization_string)
self.timer.start('rotate_psi')
operator.matrix_multiply(C_nn, psit_nG, P_ani, out_nG=kpt.psit_nG)
self.timer.stop('rotate_psi')
self.timer.stop('Orthonormalize')