def modified_cholesky_direct(M, tol=1e-5, verbose=False, cmax=20):
"""Modified cholesky decomposition of matrix.
See, e.g. [Motta17]_
Parameters
----------
M : :class:`numpy.ndarray`
Positive semi-definite, symmetric matrix.
tol : float
Accuracy desired. Optional. Default : False.
verbose : bool
If true print out convergence progress. Optional. Default : False.
cmax : int
Number of cholesky vectors to store N_gamma = cmax M.
Returns
-------
chol_vecs : :class:`numpy.ndarray`
Matrix of cholesky vectors.
"""
# matrix of residuals.
delta = numpy.copy(M.diagonal())
nchol_max = int(cmax * M.shape[0]**0.5)
# index of largest diagonal element of residual matrix.
nu = numpy.argmax(numpy.abs(delta))
delta_max = delta[nu]
if verbose:
print("# max number of cholesky vectors = %d" % nchol_max)
header = ['iteration', 'max_residual', 'time']
print(format_fixed_width_strings(header))
init = [delta_max]
print('{:17d} '.format(0) + format_fixed_width_floats(init))
# print ("# iteration %d: delta_max = %f"%(0, delta_max.real))
# Store for current approximation to input matrix.
Mapprox = numpy.zeros(M.shape[0], dtype=M.dtype)
chol_vecs = numpy.zeros((nchol_max, M.shape[0]), dtype=M.dtype)
nchol = 0
chol_vecs[0] = numpy.copy(M[:, nu]) / delta_max**0.5
while abs(delta_max) > tol:
# Update cholesky vector
start = time.time()
Mapprox += chol_vecs[nchol] * chol_vecs[nchol].conj()
delta = M.diagonal() - Mapprox
nu = numpy.argmax(numpy.abs(delta))
delta_max = numpy.abs(delta[nu])
nchol += 1
Munu0 = numpy.dot(chol_vecs[:nchol, nu].conj(), chol_vecs[:nchol, :])
chol_vecs[nchol] = (M[:, nu] - Munu0) / (delta_max)**0.5
if verbose:
step_time = time.time() - start
out = [delta_max, step_time]
print('{:17d} '.format(nchol) + format_fixed_width_floats(out))
return numpy.array(chol_vecs[:nchol])
def chunked_cholesky(mol, max_error=1e-6, verbose=False, cmax=10):
"""Modified cholesky decomposition from pyscf eris.
See, e.g. [Motta17]_
Only works for molecular systems.
Parameters
----------
mol : :class:`pyscf.mol`
pyscf mol object.
orthoAO: :class:`numpy.ndarray`
Orthogonalising matrix for AOs. (e.g., mo_coeff).
delta : float
Accuracy desired.
verbose : bool
If true print out convergence progress.
cmax : int
nchol = cmax * M, where M is the number of basis functions.
Controls buffer size for cholesky vectors.
Returns
-------
chol_vecs : :class:`numpy.ndarray`
Matrix of cholesky vectors in AO basis.
"""
nao = mol.nao_nr()
diag = numpy.zeros(nao*nao)
nchol_max = cmax * nao
# This shape is more convenient for pauxy.
chol_vecs = numpy.zeros((nchol_max, nao*nao))
# eri = numpy.zeros((nao,nao,nao,nao))
ndiag = 0
dims = [0]
nao_per_i = 0
for i in range(0,mol.nbas):
l = mol.bas_angular(i)
nc = mol.bas_nctr(i)
nao_per_i += (2*l+1)*nc
dims.append(nao_per_i)
start = time.time()
for i in range(0,mol.nbas):
shls = (i,i+1,0,mol.nbas,i,i+1,0,mol.nbas)
buf = mol.intor('int2e_sph', shls_slice=shls)
di, dk, dj, dl = buf.shape
diag[ndiag:ndiag+di*nao] = buf.reshape(di*nao,di*nao).diagonal()
ndiag += di * nao
nu = numpy.argmax(diag)
delta_max = diag[nu]
if verbose:
print(" # Generating Cholesky decomposition of ERIs."%nchol_max)
print(" # max number of cholesky vectors = %d"%nchol_max)
header = ['iteration', 'max_residual', 'time']
print(format_fixed_width_strings(header))
init = [delta_max, time.time()-start]
print('{:17d} '.format(0)+format_fixed_width_floats(init))
j = nu // nao
l = nu % nao
sj = numpy.searchsorted(dims, j)
sl = numpy.searchsorted(dims, l)
if dims[sj] != j and j != 0:
sj -= 1
if dims[sl] != l and l != 0:
sl -= 1
Mapprox = numpy.zeros(nao*nao)
# ERI[:,jl]
eri_col = mol.intor('int2e_sph',
shls_slice=(0,mol.nbas,0,mol.nbas,sj,sj+1,sl,sl+1))
cj, cl = max(j-dims[sj],0), max(l-dims[sl],0)
chol_vecs[0] = numpy.copy(eri_col[:,:,cj,cl].reshape(nao*nao)) / delta_max**0.5
nchol = 0
while abs(delta_max) > max_error:
# Update cholesky vector
start = time.time()
# M'_ii = \sum_x L_i^x L_i^x
Mapprox += chol_vecs[nchol] * chol_vecs[nchol]
# D_ii = M_ii - M'_ii
delta = diag - Mapprox
nu = numpy.argmax(numpy.abs(delta))
delta_max = numpy.abs(delta[nu])
# Compute ERI chunk.
# shls_slice computes shells of integrals as determined by the angular
# momentum of the basis function and the number of contraction
# coefficients. Need to search for AO index within this shell indexing
# scheme.
# AO index.
j = nu // nao
l = nu % nao
# Associated shell index.
sj = numpy.searchsorted(dims, j)
sl = numpy.searchsorted(dims, l)
if dims[sj] != j and j != 0:
sj -= 1
if dims[sl] != l and l != 0:
sl -= 1
# Compute ERI chunk.
eri_col = mol.intor('int2e_sph',
shls_slice=(0,mol.nbas,0,mol.nbas,sj,sj+1,sl,sl+1))
# Select correct ERI chunk from shell.
cj, cl = max(j-dims[sj],0), max(l-dims[sl],0)
Munu0 = eri_col[:,:,cj,cl].reshape(nao*nao)
# Updated residual = \sum_x L_i^x L_nu^x
R = numpy.dot(chol_vecs[:nchol+1,nu], chol_vecs[:nchol+1,:])
chol_vecs[nchol+1] = (Munu0 - R) / (delta_max)**0.5
nchol += 1
if verbose:
step_time = time.time() - start
out = [delta_max, step_time]
print('{:17d} '.format(nchol)+format_fixed_width_floats(out))
return chol_vecs[:nchol]
def run(self, comm, Xaoik, Xaolj, part, kpts, nmo_pk, nmo_max, Qi, gmap):
# Setup residual matrix.
ngs = Xaoik.shape[1]
nkpts = len(kpts)
t0 = time.clock()
residual, k1max, k2max, i1max, i2max, maxv = (self.generate_diagonal(
Xaoik, Xaolj, part, nmo_pk))
t1 = time.clock()
if part.rank == 0 and self.verbose:
print(" # Time to generate diagonal (initial residual):"
" {:.2e}".format(t1 - t0))
sys.stdout.flush()
comm.Allgather(
numpy.array([k1max, k2max, i1max, i2max, maxv],
dtype=numpy.float64), self.maxres_buff)
vmax = 0
k3, k4, i3, i4, vmax = self.find_k3k4(comm.size)
try:
done = numpy.zeros((nkpts, nkpts, nmo_max, nmo_max), numpy.int32)
maxresidual = numpy.zeros(part.maxvecs, dtype=numpy.float64)
cholvecs = numpy.zeros((part.nkk, part.nij, part.maxvecs),
dtype=numpy.complex128)
Xkl = numpy.zeros(ngs, dtype=numpy.complex128)
Xkl0 = numpy.zeros(ngs + part.maxvecs, dtype=numpy.complex128)
Vbuff = numpy.zeros(part.maxvecs, dtype=numpy.complex128)
except MemoryError:
print(" # Problems allocating memory for auxiliary structures for "
"Cholesky solver.")
done[k3, k4, i3, i4] = 1
tstart = time.clock()
if comm.rank == 0:
sys.stdout.flush()
more = True # for parallel
numv = 0
if comm.rank == 0:
header = [
"iteration", "max_residual", "total_time", "time_k3k4",
"time_comp_cholv", "time_buff"
]
if self.verbose:
print(format_fixed_width_strings(header))
while more:
t0 = time.clock()
# stop condition
if numv >= part.maxvecs:
print(" Too many vectors needed to converge. Increase maximum "
"number of vectors.")
break
kkmax = k3 * nkpts + k4
if comm.size <= nkpts:
ipr = bisect(part.kkbounds[1:comm.size + 1], kkmax)
if (kkmax >= part.kk0) & (kkmax < part.kkN):
assert (comm.rank == ipr)
else:
i34 = i3 * nmo_pk[k4] + i4
for i in range(part.nproc_pk):
ij0_, ijN_ = fair_share(nmo_max * nmo_max, part.nproc_pk,
i)
if i34 < ijN_:
ipr = k3 * part.nproc_pk + i
break
# bcast Xaolj/CV[k3,k4,i34,0:numv]
if comm.rank == ipr:
assert (((i3 * nmo_pk[k4] + i4) >= part.ij0) &
((i3 * nmo_pk[k4] + i4) < part.ijN))
Xkl0[0:ngs] = Xaolj[kkmax - part.kk0, 0:ngs,
i3 * nmo_pk[k4] + i4 - part.ij0]
Xkl0[ngs:ngs + numv] = (cholvecs[kkmax - part.kk0,
i3 * nmo_pk[k4] + i4 -
part.ij0, 0:numv])
Vbuff[0:numv] = (cholvecs[kkmax - part.kk0,
i3 * nmo_pk[k4] + i4 - part.ij0,
0:numv])
comm.Bcast(Xkl0[0:ngs + numv], root=ipr)
else:
comm.Bcast(Xkl0[0:ngs + numv], root=ipr)
Vbuff[0:numv] = Xkl0[ngs:ngs + numv]
# add new Cholesky vector
# 1. evaluate new column (ik|imax,kmax)
t1 = time.clock()
tadd = 0.0
for k in range(part.nkk):
k1 = part.n2k1[k]
k2 = part.n2k2[k]
if k3 == self.kconserv[k1, k2, k4]:
q1 = kpts[k2] - kpts[k1] + kpts[k3] - kpts[k4]
if numpy.sum(abs(q1)) > 1e-9:
t_ = time.clock()
ip = -1
for ii in range(27):
if numpy.sum(
numpy.linalg.norm(q1 - Qi[ii, :])) < 1e-12:
ip = ii
break
if ip < 0:
print(" # Could not find Q: {} {} ".format(q1, Qi))
sys.exit()
for ix in range(ngs):
Xkl[ix] = Xkl0[gmap[ip, ix]]
tadd += time.clock() - t_
else:
Xkl[0:ngs] = Xkl0[0:ngs]
n_ = min(nmo_pk[k1] * nmo_pk[k2], part.ijN) - part.ij0
cholvecs[k, 0:n_,
numv] = numpy.dot(Xaoik[k, :, 0:n_].T, Xkl.conj())
t2 = time.clock()
# 2. subtract projection along previous components
cholvecs[:, :, numv] -= numpy.dot(cholvecs[:, :, 0:numv],
Vbuff[0:numv].conj())
cholvecs[:, :, numv] /= math.sqrt(vmax)
# update residual
residual -= (cholvecs[:, :, numv] *
cholvecs[:, :, numv].conj()).real
maxv = 0
for k in range(part.nkk):
k1 = part.n2k1[k]
k2 = part.n2k2[k]
for ij in range(part.nij):
if (ij + part.ij0) >= nmo_pk[k1] * nmo_pk[k2]:
break
if abs(residual[k, ij]) > maxv:
maxv = abs(residual[k, ij])
k1max = k1
k2max = k2
i1max = (ij + part.ij0) // nmo_pk[k2]
i2max = (ij + part.ij0) % nmo_pk[k2]
t3 = time.clock()
# assemble full CV on head node
comm.Allgather(
numpy.array([k1max, k2max, i1max, i2max, maxv],
dtype=numpy.float64), self.maxres_buff)
k3, k4, i3, i4, vmax = self.find_k3k4(comm.size)
t4 = time.clock()
# only root keeps track of residual and I/O
if part.rank == 0:
maxresidual[numv] = abs(vmax)
# print and evaluate stop condition
output = [vmax, t4 - t0, t3 - t2, t2 - t1, t1 - t0]
if self.verbose:
print("{:17d} ".format(numv) +
format_fixed_width_floats(output))
# print("{:8d} {:13.8e}".format(numv, vmax))
tstart = time.clock()
if numv % 100 == 0:
sys.stdout.flush()
numv += 1
if vmax < self.gtol_chol:
more = False
if (done[k3, k4, i3, i4] > 0) and more:
print("Error in Cholesky decomposition. "
"done[imax,kmax]>0.", k3, k4, i3, i4, vmax)
sys.exit()
done[k3, k4, i3, i4] = 1
t6 = time.clock()
return cholvecs[:, :, :numv]
def run(self, comm, X, h5file):
# Unpack for convenience.
ngs = self.ngs
nmo_max = self.nmo_max
nkpts = self.nkpts
part = self.part
nmo_pk = self.nmo_pk
QKToK2 = self.QKToK2
kpts = self.kpts
# Setup residual matrix.
mem = 2 * 16 * nkpts * ngs * nmo_max**2 / 1024**3
if self.verbose and comm.rank == 0:
print(" # Approx total memory required for orbital products: "
"{:.2e} GB.".format(mem))
mem_verbose = comm.rank == 0 and self.verbose > 1
if mem_verbose:
print(" # Approx memory per MPI task for auxiliary structures.")
Xaoik = alloc_helper((part.nkk, ngs, part.nij),
dtype=numpy.complex128,
name='Xaoik',
verbose=mem_verbose)
Xaolj = alloc_helper((part.nkk, ngs, part.nij),
dtype=numpy.complex128,
name='Xaolj',
verbose=mem_verbose)
done = alloc_helper((nkpts, nmo_max, nmo_max),
numpy.int32,
name='done',
verbose=mem_verbose)
maxresidual = alloc_helper((part.maxvecs, ),
dtype=numpy.float64,
name='maxresidual',
verbose=mem_verbose)
cholvecs = alloc_helper((part.nkk, part.nij, part.maxvecs),
dtype=numpy.complex128,
name='cholvecs',
verbose=mem_verbose)
Xkl = alloc_helper((ngs, ),
dtype=numpy.complex128,
name='xkl',
verbose=mem_verbose)
Xkl0 = alloc_helper((ngs + part.maxvecs, ),
dtype=numpy.complex128,
name='xkl0',
verbose=mem_verbose)
Vbuff = alloc_helper((part.maxvecs, ),
dtype=numpy.complex128,
name='Vbuff',
verbose=mem_verbose)
num_cholvecs = alloc_helper((nkpts, ),
dtype=numpy.int32,
name='num_cholvecs',
verbose=mem_verbose)
header = [
"iteration", "max_residual", "total_time", "time_k3k4",
"time_comp_cholv", "time_buff"
]
for Q in range(nkpts):
t0 = time.clock()
if Q > self.kminus[Q]:
continue
if comm.rank == 0 and self.verbose:
print(
" # Calculating factorization for momentum: {}".format(Q))
print(" # Generating orbital products")
sys.stdout.flush()
t1 = time.clock()
maxresidual[:] = 0
done[:, :, :] = 0
cholvecs[:, :, :] = 0
start = time.time()
self.generate_orbital_products(Q, X, Xaoik, Xaolj)
if self.verbose and comm.rank == 0:
print(" # Time to generate orbital products: "
"{:13.8e}".format(time.time() - start))
residual, k1max, k2max, i1max, i2max, maxv = (
self.generate_diagonal(Q, Xaoik, Xaolj))
buff = numpy.array([k1max, k2max, i1max, i2max, maxv],
dtype=numpy.float64)
comm.Allgather(buff, self.maxres_buff)
k3, k4, i3, i4, vmax = self.find_k3k4(comm.size)
done[k3, i3, i4] = 1
tstart = time.clock()
if comm.rank == 0:
sys.stdout.flush()
cnt = 0
vmaxold = vmax
more = True # for parallel
numv = 0
while more:
t0 = time.clock()
# stop condition
if comm.rank == 0 and self.verbose:
if numv == 0:
print(format_fixed_width_strings(header))
if numv >= self.maxvecs:
print(" Too many vectors needed to converge. "
"Increase maximum number of vectors.")
break
if comm.size <= nkpts:
ipr = bisect(part.kkbounds[1:comm.size + 1], k3)
if (k3 >= part.kk0) and (k3 < part.kkN):
assert (comm.rank == ipr)
else:
i34 = i3 * nmo_pk[k4] + i4
for i in range(part.nproc_pk):
ij0_, ijN_ = fair_share(nmo_max * nmo_max,
part.nproc_pk, i)
if i34 < ijN_:
ipr = k3 * part.nproc_pk + i
break
if comm.rank == ipr:
assert ((i3 * nmo_pk[k4] + i4 >= part.ij0))
assert ((i3 * nmo_pk[k4] + i4 < part.ijN))
Xkl0[0:ngs] = Xaolj[k3 - part.kk0, 0:ngs,
i3 * nmo_pk[k4] + i4 - part.ij0]
Xkl0[ngs:ngs + numv] = cholvecs[k3 - part.kk0,
i3 * nmo_pk[k4] + i4 -
part.ij0, 0:numv]
Vbuff[0:numv] = cholvecs[k3 - part.kk0,
i3 * nmo_pk[k4] + i4 - part.ij0,
0:numv]
comm.Bcast(Xkl0[0:ngs + numv], root=ipr)
else:
comm.Bcast(Xkl0[0:ngs + numv], root=ipr)
Vbuff[0:numv] = Xkl0[ngs:ngs + numv]
# add new Cholesky vector
# 1. evaluate new column (ik|imax,kmax)
t1 = time.clock()
tadd = 0.0
for k in range(part.nkk):
k1 = k + part.kk0
k2 = QKToK2[Q][k1]
if part.ij0 > nmo_pk[k1] * nmo_pk[k2]:
continue
if numpy.sum(abs(kpts[k2] - kpts[k1] + kpts[k3] -
kpts[k4])) > 1e-9:
t_ = time.clock()
q1 = kpts[k2] - kpts[k1] + kpts[k3] - kpts[k4]
ip = -1
for ii in range(27):
if numpy.sum(
numpy.linalg.norm(q1 -
self.Qi[ii, :])) < 1e-12:
ip = ii
break
if ip < 0:
print("Could not find Q: {} {}".format(
q1, self.Qi))
sys.exit()
for ix in range(ngs):
Xkl[ix] = Xkl0[self.gmap[ip, ix]]
tadd += time.clock() - t_
else:
Xkl[0:ngs] = Xkl0[0:ngs]
n_ = min(nmo_pk[k1] * nmo_pk[k2], part.ijN) - part.ij0
cholvecs[k, 0:n_,
numv] = numpy.dot(Xaoik[k, :, 0:n_].T, Xkl.conj())
t2 = time.clock()
# 2. substract projection along previous components
cholvecs[:, :, numv] -= numpy.dot(cholvecs[:, :, 0:numv],
Vbuff[0:numv].conj())
cholvecs[:, :, numv] /= math.sqrt(vmax)
# update residual
residual -= (cholvecs[:, :, numv] *
cholvecs[:, :, numv].conj()).real
maxv = 0
k1max = -1
k2max = -1
i1max = -1
i2max = -1
for k in range(part.nkk):
k1 = k + part.kk0
k2 = self.QKToK2[Q][k1]
# if ij0 > nmo_pk[k1]*nmo_pk[k2]:
# continue
for ij in range(part.nij):
if (ij + part.ij0) >= nmo_pk[k1] * nmo_pk[k2]:
break
if abs(residual[k, ij]) > maxv:
maxv = abs(residual[k, ij])
k1max = k1
k2max = k2
i1max = (ij + part.ij0) // nmo_pk[k2]
i2max = (ij + part.ij0) % nmo_pk[k2]
t3 = time.clock()
# assemble full CV on head node
buff = numpy.array([k1max, k2max, i1max, i2max, maxv],
dtype=numpy.float64)
comm.Allgather(buff, self.maxres_buff)
k3, k4, i3, i4, vmax = self.find_k3k4(comm.size)
t4 = time.clock()
# only root keeps track of residual and I/O
if comm.rank == 0:
maxresidual[numv] = abs(vmax)
# print and evaluate stop condition
output = [vmax, t4 - t0, t3 - t2, t2 - t1, t1 - t0]
if self.verbose:
print("{:17d} ".format(numv) +
format_fixed_width_floats(output))
tstart = time.clock()
if numv % 100 == 0:
sys.stdout.flush()
numv += 1
if vmax < self.gtol_chol:
more = False
if done[k3, i3, i4] > 0 and more:
# TODO: What is this?
print(
"Error in Cholesky decomposition. "
"done[imax,kmax]>0.", k3, k4, i3, i4, vmax)
sys.exit()
done[k3, i3, i4] = 1
t6 = time.clock()
comm.barrier()
num_cholvecs[Q] = numv
if h5file.phdf or comm.rank == 0:
LQ = h5file.grp_v2.create_dataset(
"L" + str(Q), (nkpts, nmo_max * nmo_max * numv, 2),
dtype=numpy.float64)
# cholvecs[nkk,nij,maxvecs]
for kk in range(part.nkk):
T_ = to_qmcpack_complex(cholvecs[kk, :, 0:numv].copy())
T_ = numpy.reshape(T_, (-1, 2)) / math.sqrt(nkpts * 1.0)
LQ[kk + part.kk0, part.ij0 * numv:part.ijN * numv, :] = T_
T_ = None
else:
Ldim = h5file.grp_v2.create_dataset(
"Ldim" + str(Q),
data=numpy.array([
part.nkk, part.nij, part.kk0, part.ij0, part.ijN, numv
],
dtype=numpy.int32))
LQ = h5file.grp_v2.create_dataset(
"L" + str(Q), (part.nkk, part.nij * numv, 2),
dtype=numpy.float64)
# cholvecs[nkk,nij,maxvecs]
for kk in range(part.nkk):
T_ = to_qmcpack_complex(cholvecs[kk, :, 0:numv].copy())
T_ = numpy.reshape(T_, (-1, 2)) / math.sqrt(nkpts * 1.0)
LQ[kk, :, :] = T_
T_ = None
comm.barrier()
h5file.grp.create_dataset("NCholPerKP", data=num_cholvecs)
comm.barrier()