"""
Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG).
References
----------
.. [1] A. V. Knyazev (2001),
Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method.
SIAM Journal on Scientific Computing 23, no. 2,
pp. 517-541. :doi:`10.1137/S1064827500366124`
.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov (2007),
Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX)
in hypre and PETSc. :arxiv:`0705.2626`
.. [3] A. V. Knyazev's C and MATLAB implementations:
https://github.com/lobpcg/blopex
"""
import warnings
import numpy as np
from scipy.linalg import (inv, eigh, cho_factor, cho_solve,
cholesky, LinAlgError)
from scipy.sparse.linalg import LinearOperator
from scipy.sparse import issparse
__all__ = ["lobpcg"]
def _report_nonhermitian(M, name):
"""
Report if `M` is not a Hermitian matrix given its type.
"""
from scipy.linalg import norm
md = M - M.T.conj()
nmd = norm(md, 1)
tol = 10 * np.finfo(M.dtype).eps
tol = max(tol, tol * norm(M, 1))
if nmd > tol:
warnings.warn(
f"Matrix {name} of the type {M.dtype} is not Hermitian: "
f"condition: {nmd} < {tol} fails.",
UserWarning, stacklevel=4
)
def _as2d(ar):
"""
If the input array is 2D return it, if it is 1D, append a dimension,
making it a column vector.
"""
if ar.ndim == 2:
return ar
else: # Assume 1!
aux = np.asarray(ar)
aux.shape = (ar.shape[0], 1)
return aux
def _makeMatMat(m):
if m is None:
return None
elif callable(m):
return lambda v: m(v)
else:
return lambda v: m @ v
def _matmul_inplace(x, y, verbosityLevel=0):
"""Perform 'np.matmul' in-place if possible.
If some sufficient conditions for inplace matmul are met, do so.
Otherwise try inplace update and fall back to overwrite if that fails.
"""
if x.flags["CARRAY"] and x.shape[1] == y.shape[1] and x.dtype == y.dtype:
# conditions where we can guarantee that inplace updates will work;
# i.e. x is not a view/slice, x & y have compatible dtypes, and the
# shape of the result of x @ y matches the shape of x.
np.matmul(x, y, out=x)
else:
# ideally, we'd have an exhaustive list of conditions above when
# inplace updates are possible; since we don't, we opportunistically
# try if it works, and fall back to overwriting if necessary
try:
np.matmul(x, y, out=x)
except Exception:
if verbosityLevel:
warnings.warn(
"Inplace update of x = x @ y failed, "
"x needs to be overwritten.",
UserWarning, stacklevel=3
)
x = x @ y
return x
def _applyConstraints(blockVectorV, factYBY, blockVectorBY, blockVectorY):
"""Changes blockVectorV in-place."""
YBV = blockVectorBY.T.conj() @ blockVectorV
tmp = cho_solve(factYBY, YBV)
blockVectorV -= blockVectorY @ tmp
def _b_orthonormalize(B, blockVectorV, blockVectorBV=None,
verbosityLevel=0):
"""in-place B-orthonormalize the given block vector using Cholesky."""
if blockVectorBV is None:
if B is None:
blockVectorBV = blockVectorV
else:
try:
blockVectorBV = B(blockVectorV)
except Exception as e:
if verbosityLevel:
warnings.warn(
f"Secondary MatMul call failed with error\n"
f"{e}\n",
UserWarning, stacklevel=3
)
return None, None, None
if blockVectorBV.shape != blockVectorV.shape:
raise ValueError(
f"The shape {blockVectorV.shape} "
f"of the orthogonalized matrix not preserved\n"
f"and changed to {blockVectorBV.shape} "
f"after multiplying by the secondary matrix.\n"
)
VBV = blockVectorV.T.conj() @ blockVectorBV
try:
# VBV is a Cholesky factor from now on...
VBV = cholesky(VBV, overwrite_a=True)
VBV = inv(VBV, overwrite_a=True)
blockVectorV = _matmul_inplace(
blockVectorV, VBV,
verbosityLevel=verbosityLevel
)
if B is not None:
blockVectorBV = _matmul_inplace(
blockVectorBV, VBV,
verbosityLevel=verbosityLevel
)
return blockVectorV, blockVectorBV, VBV
except LinAlgError:
if verbosityLevel:
warnings.warn(
"Cholesky has failed.",
UserWarning, stacklevel=3
)
return None, None, None
def _get_indx(_lambda, num, largest):
"""Get `num` indices into `_lambda` depending on `largest` option."""
ii = np.argsort(_lambda)
if largest:
ii = ii[:-num - 1:-1]
else:
ii = ii[:num]
return ii
def _handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel):
if verbosityLevel:
_report_nonhermitian(gramA, "gramA")
_report_nonhermitian(gramB, "gramB")
def lobpcg(
A,
X,
B=None,
M=None,
Y=None,
tol=None,
maxiter=None,
largest=True,
verbosityLevel=0,
retLambdaHistory=False,
retResidualNormsHistory=False,
restartControl=20,
):
"""Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG).
LOBPCG is a preconditioned eigensolver for large real symmetric and complex
Hermitian definite generalized eigenproblems.
Parameters
----------
A : {sparse matrix, ndarray, LinearOperator, callable object}
The Hermitian linear operator of the problem, usually given by a
sparse matrix. Often called the "stiffness matrix".
X : ndarray, float32 or float64
Initial approximation to the ``k`` eigenvectors (non-sparse).
If `A` has ``shape=(n,n)`` then `X` must have ``shape=(n,k)``.
B : {sparse matrix, ndarray, LinearOperator, callable object}
Optional. By default ``B = None``, which is equivalent to identity.
The right hand side operator in a generalized eigenproblem if present.
Often called the "mass matrix". Must be Hermitian positive definite.
M : {sparse matrix, ndarray, LinearOperator, callable object}
Optional. By default ``M = None``, which is equivalent to identity.
Preconditioner aiming to accelerate convergence.
Y : ndarray, float32 or float64, default: None
An ``n-by-sizeY`` ndarray of constraints with ``sizeY < n``.
The iterations will be performed in the ``B``-orthogonal complement
of the column-space of `Y`. `Y` must be full rank if present.
tol : scalar, optional
The default is ``tol=n*sqrt(eps)``.
Solver tolerance for the stopping criterion.
maxiter : int, default: 20
Maximum number of iterations.
largest : bool, default: True
When True, solve for the largest eigenvalues, otherwise the smallest.
verbosityLevel : int, optional
By default ``verbosityLevel=0`` no output.
Controls the solver standard/screen output.
retLambdaHistory : bool, default: False
Whether to return iterative eigenvalue history.
retResidualNormsHistory : bool, default: False
Whether to return iterative history of residual norms.
restartControl : int, optional.
Iterations restart if the residuals jump ``2**restartControl`` times
compared to the smallest recorded in ``retResidualNormsHistory``.
The default is ``restartControl=20``, making the restarts rare for
backward compatibility.
Returns
-------
lambda : ndarray of the shape ``(k, )``.
Array of ``k`` approximate eigenvalues.
v : ndarray of the same shape as ``X.shape``.
An array of ``k`` approximate eigenvectors.
lambdaHistory : ndarray, optional.
The eigenvalue history, if `retLambdaHistory` is ``True``.
ResidualNormsHistory : ndarray, optional.
The history of residual norms, if `retResidualNormsHistory`
is ``True``.
Notes
-----
The iterative loop runs ``maxit=maxiter`` (20 if ``maxit=None``)
iterations at most and finishes earlier if the tolerance is met.
Breaking backward compatibility with the previous version, LOBPCG
now returns the block of iterative vectors with the best accuracy rather
than the last one iterated, as a cure for possible divergence.
If ``X.dtype == np.float32`` and user-provided operations/multiplications
by `A`, `B`, and `M` all preserve the ``np.float32`` data type,
all the calculations and the output are in ``np.float32``.
The size of the iteration history output equals to the number of the best
(limited by `maxit`) iterations plus 3: initial, final, and postprocessing.
If both `retLambdaHistory` and `retResidualNormsHistory` are ``True``,
the return tuple has the following format
``(lambda, V, lambda history, residual norms history)``.
In the following ``n`` denotes the matrix size and ``k`` the number
of required eigenvalues (smallest or largest).
The LOBPCG code internally solves eigenproblems of the size ``3k`` on every
iteration by calling the dense eigensolver `eigh`, so if ``k`` is not
small enough compared to ``n``, it makes no sense to call the LOBPCG code.
Moreover, if one calls the LOBPCG algorithm for ``5k > n``, it would likely
break internally, so the code calls the standard function `eigh` instead.
It is not that ``n`` should be large for the LOBPCG to work, but rather the
ratio ``n / k`` should be large. It you call LOBPCG with ``k=1``
and ``n=10``, it works though ``n`` is small. The method is intended
for extremely large ``n / k``.
The convergence speed depends basically on three factors:
1. Quality of the initial approximations `X` to the seeking eigenvectors.
Randomly distributed around the origin vectors work well if no better
choice is known.
2. Relative separation of the desired eigenvalues from the rest
of the eigenvalues. One can vary ``k`` to improve the separation.
3. Proper preconditioning to shrink the spectral spread.
For example, a rod vibration test problem (under tests
directory) is ill-conditioned for large ``n``, so convergence will be
slow, unless efficient preconditioning is used. For this specific
problem, a good simple preconditioner function would be a linear solve
for `A`, which is easy to code since `A` is tridiagonal.
References
----------
.. [1] A. V. Knyazev (2001),
Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method.
SIAM Journal on Scientific Computing 23, no. 2,
pp. 517-541. :doi:`10.1137/S1064827500366124`
.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov
(2007), Block Locally Optimal Preconditioned Eigenvalue Xolvers
(BLOPEX) in hypre and PETSc. :arxiv:`0705.2626`
.. [3] A. V. Knyazev's C and MATLAB implementations:
https://github.com/lobpcg/blopex
Examples
--------
Our first example is minimalistic - find the largest eigenvalue of
a diagonal matrix by solving the non-generalized eigenvalue problem
``A x = lambda x`` without constraints or preconditioning.
>>> import numpy as np
>>> from scipy.sparse import spdiags
>>> from scipy.sparse.linalg import LinearOperator, aslinearoperator
>>> from scipy.sparse.linalg import lobpcg
The square matrix size is
>>> n = 100
and its diagonal entries are 1, ..., 100 defined by
>>> vals = np.arange(1, n + 1).astype(np.int16)
The first mandatory input parameter in this test is
the sparse diagonal matrix `A`
of the eigenvalue problem ``A x = lambda x`` to solve.
>>> A = spdiags(vals, 0, n, n)
>>> A = A.astype(np.int16)
>>> A.toarray()
array([[ 1, 0, 0, ..., 0, 0, 0],
[ 0, 2, 0, ..., 0, 0, 0],
[ 0, 0, 3, ..., 0, 0, 0],
...,
[ 0, 0, 0, ..., 98, 0, 0],
[ 0, 0, 0, ..., 0, 99, 0],
[ 0, 0, 0, ..., 0, 0, 100]], dtype=int16)
The second mandatory input parameter `X` is a 2D array with the
row dimension determining the number of requested eigenvalues.
`X` is an initial guess for targeted eigenvectors.
`X` must have linearly independent columns.
If no initial approximations available, randomly oriented vectors
commonly work best, e.g., with components normally distributed
around zero or uniformly distributed on the interval [-1 1].
Setting the initial approximations to dtype ``np.float32``
forces all iterative values to dtype ``np.float32`` speeding up
the run while still allowing accurate eigenvalue computations.
>>> k = 1
>>> rng = np.random.default_rng()
>>> X = rng.normal(size=(n, k))
>>> X = X.astype(np.float32)
>>> eigenvalues, _ = lobpcg(A, X, maxiter=60)
>>> eigenvalues
array([100.])
>>> eigenvalues.dtype
dtype('float32')
`lobpcg` needs only access the matrix product with `A` rather
then the matrix itself. Since the matrix `A` is diagonal in
this example, one can write a function of the matrix product
``A @ X`` using the diagonal values ``vals`` only, e.g., by
element-wise multiplication with broadcasting in the lambda-function
>>> A_lambda = lambda X: vals[:, np.newaxis] * X
or the regular function
>>> def A_matmat(X):
... return vals[:, np.newaxis] * X
and use the handle to one of these callables as an input
>>> eigenvalues, _ = lobpcg(A_lambda, X, maxiter=60)
>>> eigenvalues
array([100.])
>>> eigenvalues, _ = lobpcg(A_matmat, X, maxiter=60)
>>> eigenvalues
array([100.])
The traditional callable `LinearOperator` is no longer
necessary but still supported as the input to `lobpcg`.
Specifying ``matmat=A_matmat`` explicitly improves performance.
>>> A_lo = LinearOperator((n, n), matvec=A_matmat, matmat=A_matmat, dtype=np.int16)
>>> eigenvalues, _ = lobpcg(A_lo, X, maxiter=80)
>>> eigenvalues
array([100.])
The least efficient callable option is `aslinearoperator`:
>>> eigenvalues, _ = lobpcg(aslinearoperator(A), X, maxiter=80)
>>> eigenvalues
array([100.])
We now switch to computing the three smallest eigenvalues specifying
>>> k = 3
>>> X = np.random.default_rng().normal(size=(n, k))
and ``largest=False`` parameter
>>> eigenvalues, _ = lobpcg(A, X, largest=False, maxiter=80)
>>> print(eigenvalues)
[1. 2. 3.]
The next example illustrates computing 3 smallest eigenvalues of
the same matrix `A` given by the function handle ``A_matmat`` but
with constraints and preconditioning.
Constraints - an optional input parameter is a 2D array comprising
of column vectors that the eigenvectors must be orthogonal to
>>> Y = np.eye(n, 3)
The preconditioner acts as the inverse of `A` in this example, but
in the reduced precision ``np.float32`` even though the initial `X`
and thus all iterates and the output are in full ``np.float64``.
>>> inv_vals = 1./vals
>>> inv_vals = inv_vals.astype(np.float32)
>>> M = lambda X: inv_vals[:, np.newaxis] * X
Let us now solve the eigenvalue problem for the matrix `A` first
without preconditioning requesting 80 iterations
>>> eigenvalues, _ = lobpcg(A_matmat, X, Y=Y, largest=False, maxiter=80)
>>> eigenvalues
array([4., 5., 6.])
>>> eigenvalues.dtype
dtype('float64')
With preconditioning we need only 20 iterations from the same `X`
>>> eigenvalues, _ = lobpcg(A_matmat, X, Y=Y, M=M, largest=False, maxiter=20)
>>> eigenvalues
array([4., 5., 6.])
Note that the vectors passed in `Y` are the eigenvectors of the 3
smallest eigenvalues. The results returned above are orthogonal to those.
The primary matrix `A` may be indefinite, e.g., after shifting
``vals`` by 50 from 1, ..., 100 to -49, ..., 50, we still can compute
the 3 smallest or largest eigenvalues.
>>> vals = vals - 50
>>> X = rng.normal(size=(n, k))
>>> eigenvalues, _ = lobpcg(A_matmat, X, largest=False, maxiter=99)
>>> eigenvalues
array([-49., -48., -47.])
>>> eigenvalues, _ = lobpcg(A_matmat, X, largest=True, maxiter=99)
>>> eigenvalues
array([50., 49., 48.])
"""
blockVectorX = X
bestblockVectorX = blockVectorX
blockVectorY = Y
residualTolerance = tol
if maxiter is None:
maxiter = 20
bestIterationNumber = maxiter
sizeY = 0
if blockVectorY is not None:
if len(blockVectorY.shape) != 2:
warnings.warn(
f"Expected rank-2 array for argument Y, instead got "
f"{len(blockVectorY.shape)}, "
f"so ignore it and use no constraints.",
UserWarning, stacklevel=2
)
blockVectorY = None
else:
sizeY = blockVectorY.shape[1]
# Block size.
if blockVectorX is None:
raise ValueError("The mandatory initial matrix X cannot be None")
if len(blockVectorX.shape) != 2:
raise ValueError("expected rank-2 array for argument X")
n, sizeX = blockVectorX.shape
# Data type of iterates, determined by X, must be inexact
if not np.issubdtype(blockVectorX.dtype, np.inexact):
warnings.warn(
f"Data type for argument X is {blockVectorX.dtype}, "
f"which is not inexact, so casted to np.float32.",
UserWarning, stacklevel=2
)
blockVectorX = np.asarray(blockVectorX, dtype=np.float32)
if retLambdaHistory:
lambdaHistory = np.zeros((maxiter + 3, sizeX),
dtype=blockVectorX.dtype)
if retResidualNormsHistory:
residualNormsHistory = np.zeros((maxiter + 3, sizeX),
dtype=blockVectorX.dtype)
if verbosityLevel:
aux = "Solving "
if B is None:
aux += "standard"
else:
aux += "generalized"
aux += " eigenvalue problem with"
if M is None:
aux += "out"
aux += " preconditioning\n\n"
aux += "matrix size %d\n" % n
aux += "block size %d\n\n" % sizeX
if blockVectorY is None:
aux += "No constraints\n\n"
else:
if sizeY > 1:
aux += "%d constraints\n\n" % sizeY
else:
aux += "%d constraint\n\n" % sizeY
print(aux)
if (n - sizeY) < (5 * sizeX):
warnings.warn(
f"The problem size {n} minus the constraints size {sizeY} "
f"is too small relative to the block size {sizeX}. "
f"Using a dense eigensolver instead of LOBPCG iterations."
f"No output of the history of the iterations.",
UserWarning, stacklevel=2
)
sizeX = min(sizeX, n)
if blockVectorY is not None:
raise NotImplementedError(
"The dense eigensolver does not support constraints."
)
# Define the closed range of indices of eigenvalues to return.
if largest:
eigvals = (n - sizeX, n - 1)
else:
eigvals = (0, sizeX - 1)
try:
if isinstance(A, LinearOperator):
A = A(np.eye(n, dtype=int))
elif callable(A):
A = A(np.eye(n, dtype=int))
if A.shape != (n, n):
raise ValueError(
f"The shape {A.shape} of the primary matrix\n"
f"defined by a callable object is wrong.\n"
)
elif issparse(A):
A = A.toarray()
else:
A = np.asarray(A)
except Exception as e:
raise Exception(
f"Primary MatMul call failed with error\n"
f"{e}\n")
if B is not None:
try:
if isinstance(B, LinearOperator):
B = B(np.eye(n, dtype=int))
elif callable(B):
B = B(np.eye(n, dtype=int))
if B.shape != (n, n):
raise ValueError(
f"The shape {B.shape} of the secondary matrix\n"
f"defined by a callable object is wrong.\n"
)
elif issparse(B):
B = B.toarray()
else:
B = np.asarray(B)
except Exception as e:
raise Exception(
f"Secondary MatMul call failed with error\n"
f"{e}\n")
try:
vals, vecs = eigh(A,
B,
subset_by_index=eigvals,
check_finite=False)
if largest:
# Reverse order to be compatible with eigs() in 'LM' mode.
vals = vals[::-1]
vecs = vecs[:, ::-1]
return vals, vecs
except Exception as e:
raise Exception(
f"Dense eigensolver failed with error\n"
f"{e}\n"
)
if (residualTolerance is None) or (residualTolerance <= 0.0):
residualTolerance = np.sqrt(np.finfo(blockVectorX.dtype).eps) * n
A = _makeMatMat(A)
B = _makeMatMat(B)
M = _makeMatMat(M)
# Apply constraints to X.
if blockVectorY is not None:
if B is not None:
blockVectorBY = B(blockVectorY)
if blockVectorBY.shape != blockVectorY.shape:
raise ValueError(
f"The shape {blockVectorY.shape} "
f"of the constraint not preserved\n"
f"and changed to {blockVectorBY.shape} "
f"after multiplying by the secondary matrix.\n"
)
else:
blockVectorBY = blockVectorY
# gramYBY is a dense array.
gramYBY = blockVectorY.T.conj() @ blockVectorBY
try:
# gramYBY is a Cholesky factor from now on...
gramYBY = cho_factor(gramYBY, overwrite_a=True)
except LinAlgError as e:
raise ValueError("Linearly dependent constraints") from e
_applyConstraints(blockVectorX, gramYBY, blockVectorBY, blockVectorY)
##
# B-orthonormalize X.
blockVectorX, blockVectorBX, _ = _b_orthonormalize(
B, blockVectorX, verbosityLevel=verbosityLevel)
if blockVectorX is None:
raise ValueError("Linearly dependent initial approximations")
##
# Compute the initial Ritz vectors: solve the eigenproblem.
blockVectorAX = A(blockVectorX)
if blockVectorAX.shape != blockVectorX.shape:
raise ValueError(
f"The shape {blockVectorX.shape} "
f"of the initial approximations not preserved\n"
f"and changed to {blockVectorAX.shape} "
f"after multiplying by the primary matrix.\n"
)
gramXAX = blockVectorX.T.conj() @ blockVectorAX
_lambda, eigBlockVector = eigh(gramXAX, check_finite=False)
ii = _get_indx(_lambda, sizeX, largest)
_lambda = _lambda[ii]
if retLambdaHistory:
lambdaHistory[0, :] = _lambda
eigBlockVector = np.asarray(eigBlockVector[:, ii])
blockVectorX = _matmul_inplace(
blockVectorX, eigBlockVector,
verbosityLevel=verbosityLevel
)
blockVectorAX = _matmul_inplace(
blockVectorAX, eigBlockVector,
verbosityLevel=verbosityLevel
)
if B is not None:
blockVectorBX = _matmul_inplace(
blockVectorBX, eigBlockVector,
verbosityLevel=verbosityLevel
)
##
# Active index set.
activeMask = np.ones((sizeX,), dtype=bool)
##
# Main iteration loop.
blockVectorP = None # set during iteration
blockVectorAP = None
blockVectorBP = None
smallestResidualNorm = np.abs(np.finfo(blockVectorX.dtype).max)
iterationNumber = -1
restart = True
forcedRestart = False
explicitGramFlag = False
while iterationNumber < maxiter:
iterationNumber += 1
if B is not None:
aux = blockVectorBX * _lambda[np.newaxis, :]
else:
aux = blockVectorX * _lambda[np.newaxis, :]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
residualNorms = np.sqrt(np.abs(aux))
if retResidualNormsHistory:
residualNormsHistory[iterationNumber, :] = residualNorms
residualNorm = np.sum(np.abs(residualNorms)) / sizeX
if residualNorm < smallestResidualNorm:
smallestResidualNorm = residualNorm
bestIterationNumber = iterationNumber
bestblockVectorX = blockVectorX
elif residualNorm > 2**restartControl * smallestResidualNorm:
forcedRestart = True
blockVectorAX = A(blockVectorX)
if blockVectorAX.shape != blockVectorX.shape:
raise ValueError(
f"The shape {blockVectorX.shape} "
f"of the restarted iterate not preserved\n"
f"and changed to {blockVectorAX.shape} "
f"after multiplying by the primary matrix.\n"
)
if B is not None:
blockVectorBX = B(blockVectorX)
if blockVectorBX.shape != blockVectorX.shape:
raise ValueError(
f"The shape {blockVectorX.shape} "
f"of the restarted iterate not preserved\n"
f"and changed to {blockVectorBX.shape} "
f"after multiplying by the secondary matrix.\n"
)
ii = np.where(residualNorms > residualTolerance, True, False)
activeMask = activeMask & ii
currentBlockSize = activeMask.sum()
if verbosityLevel:
print(f"iteration {iterationNumber}")
print(f"current block size: {currentBlockSize}")
print(f"eigenvalue(s):\n{_lambda}")
print(f"residual norm(s):\n{residualNorms}")
if currentBlockSize == 0:
break
activeBlockVectorR = _as2d(blockVectorR[:, activeMask])
if iterationNumber > 0:
activeBlockVectorP = _as2d(blockVectorP[:, activeMask])
activeBlockVectorAP = _as2d(blockVectorAP[:, activeMask])
if B is not None:
activeBlockVectorBP = _as2d(blockVectorBP[:, activeMask])
if M is not None:
# Apply preconditioner T to the active residuals.
activeBlockVectorR = M(activeBlockVectorR)
##
# Apply constraints to the preconditioned residuals.
if blockVectorY is not None:
_applyConstraints(activeBlockVectorR,
gramYBY,
blockVectorBY,
blockVectorY)
##
# B-orthogonalize the preconditioned residuals to X.
if B is not None:
activeBlockVectorR = activeBlockVectorR - (
blockVectorX @
(blockVectorBX.T.conj() @ activeBlockVectorR)
)
else:
activeBlockVectorR = activeBlockVectorR - (
blockVectorX @
(blockVectorX.T.conj() @ activeBlockVectorR)
)
##
# B-orthonormalize the preconditioned residuals.
aux = _b_orthonormalize(
B, activeBlockVectorR, verbosityLevel=verbosityLevel)
activeBlockVectorR, activeBlockVectorBR, _ = aux
if activeBlockVectorR is None:
warnings.warn(
f"Failed at iteration {iterationNumber} with accuracies "
f"{residualNorms}\n not reaching the requested "
f"tolerance {residualTolerance}.",
UserWarning, stacklevel=2
)
break
activeBlockVectorAR = A(activeBlockVectorR)
if iterationNumber > 0:
if B is not None:
aux = _b_orthonormalize(
B, activeBlockVectorP, activeBlockVectorBP,
verbosityLevel=verbosityLevel
)
activeBlockVectorP, activeBlockVectorBP, invR = aux
else:
aux = _b_orthonormalize(B, activeBlockVectorP,
verbosityLevel=verbosityLevel)
activeBlockVectorP, _, invR = aux
# Function _b_orthonormalize returns None if Cholesky fails
if activeBlockVectorP is not None:
activeBlockVectorAP = _matmul_inplace(
activeBlockVectorAP, invR,
verbosityLevel=verbosityLevel
)
restart = forcedRestart
else:
restart = True
##
# Perform the Rayleigh Ritz Procedure:
# Compute symmetric Gram matrices:
if activeBlockVectorAR.dtype == "float32":
myeps = 1
else:
myeps = np.sqrt(np.finfo(activeBlockVectorR.dtype).eps)
if residualNorms.max() > myeps and not explicitGramFlag:
explicitGramFlag = False
else:
# Once explicitGramFlag, forever explicitGramFlag.
explicitGramFlag = True
# Shared memory assignments to simplify the code
if B is None:
blockVectorBX = blockVectorX
activeBlockVectorBR = activeBlockVectorR
if not restart:
activeBlockVectorBP = activeBlockVectorP
# Common submatrices:
gramXAR = np.dot(blockVectorX.T.conj(), activeBlockVectorAR)
gramRAR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR)
gramDtype = activeBlockVectorAR.dtype
if explicitGramFlag:
gramRAR = (gramRAR + gramRAR.T.conj()) / 2
gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
gramXAX = (gramXAX + gramXAX.T.conj()) / 2
gramXBX = np.dot(blockVectorX.T.conj(), blockVectorBX)
gramRBR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBR)
gramXBR = np.dot(blockVectorX.T.conj(), activeBlockVectorBR)
else:
gramXAX = np.diag(_lambda).astype(gramDtype)
gramXBX = np.eye(sizeX, dtype=gramDtype)
gramRBR = np.eye(currentBlockSize, dtype=gramDtype)
gramXBR = np.zeros((sizeX, currentBlockSize), dtype=gramDtype)
if not restart:
gramXAP = np.dot(blockVectorX.T.conj(), activeBlockVectorAP)
gramRAP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP)
gramPAP = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP)
gramXBP = np.dot(blockVectorX.T.conj(), activeBlockVectorBP)
gramRBP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBP)
if explicitGramFlag:
gramPAP = (gramPAP + gramPAP.T.conj()) / 2
gramPBP = np.dot(activeBlockVectorP.T.conj(),
activeBlockVectorBP)
else:
gramPBP = np.eye(currentBlockSize, dtype=gramDtype)
gramA = np.block(
[
[gramXAX, gramXAR, gramXAP],
[gramXAR.T.conj(), gramRAR, gramRAP],
[gramXAP.T.conj(), gramRAP.T.conj(), gramPAP],
]
)
gramB = np.block(
[
[gramXBX, gramXBR, gramXBP],
[gramXBR.T.conj(), gramRBR, gramRBP],
[gramXBP.T.conj(), gramRBP.T.conj(), gramPBP],
]
)
_handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel)
try:
_lambda, eigBlockVector = eigh(gramA,
gramB,
check_finite=False)
except LinAlgError as e:
# raise ValueError("eigh failed in lobpcg iterations") from e
if verbosityLevel:
warnings.warn(
f"eigh failed at iteration {iterationNumber} \n"
f"with error {e} causing a restart.\n",
UserWarning, stacklevel=2
)
# try again after dropping the direction vectors P from RR
restart = True
if restart:
gramA = np.block([[gramXAX, gramXAR], [gramXAR.T.conj(), gramRAR]])
gramB = np.block([[gramXBX, gramXBR], [gramXBR.T.conj(), gramRBR]])
_handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel)
try:
_lambda, eigBlockVector = eigh(gramA,
gramB,
check_finite=False)
except LinAlgError as e:
# raise ValueError("eigh failed in lobpcg iterations") from e
warnings.warn(
f"eigh failed at iteration {iterationNumber} with error\n"
f"{e}\n",
UserWarning, stacklevel=2
)
break
ii = _get_indx(_lambda, sizeX, largest)
_lambda = _lambda[ii]
eigBlockVector = eigBlockVector[:, ii]
if retLambdaHistory:
lambdaHistory[iterationNumber + 1, :] = _lambda
# Compute Ritz vectors.
if B is not None:
if not restart:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:
sizeX + currentBlockSize]
eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
pp += np.dot(activeBlockVectorP, eigBlockVectorP)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
app += np.dot(activeBlockVectorAP, eigBlockVectorP)
bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
bpp += np.dot(activeBlockVectorBP, eigBlockVectorP)
else:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
blockVectorBX = np.dot(blockVectorBX, eigBlockVectorX) + bpp
blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp
else:
if not restart:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:
sizeX + currentBlockSize]
eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
pp += np.dot(activeBlockVectorP, eigBlockVectorP)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
app += np.dot(activeBlockVectorAP, eigBlockVectorP)
else:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
blockVectorP, blockVectorAP = pp, app
if B is not None:
aux = blockVectorBX * _lambda[np.newaxis, :]
else:
aux = blockVectorX * _lambda[np.newaxis, :]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
residualNorms = np.sqrt(np.abs(aux))
# Use old lambda in case of early loop exit.
if retLambdaHistory:
lambdaHistory[iterationNumber + 1, :] = _lambda
if retResidualNormsHistory:
residualNormsHistory[iterationNumber + 1, :] = residualNorms
residualNorm = np.sum(np.abs(residualNorms)) / sizeX
if residualNorm < smallestResidualNorm:
smallestResidualNorm = residualNorm
bestIterationNumber = iterationNumber + 1
bestblockVectorX = blockVectorX
if np.max(np.abs(residualNorms)) > residualTolerance:
warnings.warn(
f"Exited at iteration {iterationNumber} with accuracies \n"
f"{residualNorms}\n"
f"not reaching the requested tolerance {residualTolerance}.\n"
f"Use iteration {bestIterationNumber} instead with accuracy \n"
f"{smallestResidualNorm}.\n",
UserWarning, stacklevel=2
)
if verbosityLevel:
print(f"Final iterative eigenvalue(s):\n{_lambda}")
print(f"Final iterative residual norm(s):\n{residualNorms}")
blockVectorX = bestblockVectorX
# Making eigenvectors "exactly" satisfy the blockVectorY constrains
if blockVectorY is not None:
_applyConstraints(blockVectorX,
gramYBY,
blockVectorBY,
blockVectorY)
# Making eigenvectors "exactly" othonormalized by final "exact" RR
blockVectorAX = A(blockVectorX)
if blockVectorAX.shape != blockVectorX.shape:
raise ValueError(
f"The shape {blockVectorX.shape} "
f"of the postprocessing iterate not preserved\n"
f"and changed to {blockVectorAX.shape} "
f"after multiplying by the primary matrix.\n"
)
gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
blockVectorBX = blockVectorX
if B is not None:
blockVectorBX = B(blockVectorX)
if blockVectorBX.shape != blockVectorX.shape:
raise ValueError(
f"The shape {blockVectorX.shape} "
f"of the postprocessing iterate not preserved\n"
f"and changed to {blockVectorBX.shape} "
f"after multiplying by the secondary matrix.\n"
)
gramXBX = np.dot(blockVectorX.T.conj(), blockVectorBX)
_handle_gramA_gramB_verbosity(gramXAX, gramXBX, verbosityLevel)
gramXAX = (gramXAX + gramXAX.T.conj()) / 2
gramXBX = (gramXBX + gramXBX.T.conj()) / 2
try:
_lambda, eigBlockVector = eigh(gramXAX,
gramXBX,
check_finite=False)
except LinAlgError as e:
raise ValueError("eigh has failed in lobpcg postprocessing") from e
ii = _get_indx(_lambda, sizeX, largest)
_lambda = _lambda[ii]
eigBlockVector = np.asarray(eigBlockVector[:, ii])
blockVectorX = np.dot(blockVectorX, eigBlockVector)
blockVectorAX = np.dot(blockVectorAX, eigBlockVector)
if B is not None:
blockVectorBX = np.dot(blockVectorBX, eigBlockVector)
aux = blockVectorBX * _lambda[np.newaxis, :]
else:
aux = blockVectorX * _lambda[np.newaxis, :]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
residualNorms = np.sqrt(np.abs(aux))
if retLambdaHistory:
lambdaHistory[bestIterationNumber + 1, :] = _lambda
if retResidualNormsHistory:
residualNormsHistory[bestIterationNumber + 1, :] = residualNorms
if retLambdaHistory:
lambdaHistory = lambdaHistory[
: bestIterationNumber + 2, :]
if retResidualNormsHistory:
residualNormsHistory = residualNormsHistory[
: bestIterationNumber + 2, :]
if np.max(np.abs(residualNorms)) > residualTolerance:
warnings.warn(
f"Exited postprocessing with accuracies \n"
f"{residualNorms}\n"
f"not reaching the requested tolerance {residualTolerance}.",
UserWarning, stacklevel=2
)
if verbosityLevel:
print(f"Final postprocessing eigenvalue(s):\n{_lambda}")
print(f"Final residual norm(s):\n{residualNorms}")
if retLambdaHistory:
lambdaHistory = np.vsplit(lambdaHistory, np.shape(lambdaHistory)[0])
lambdaHistory = [np.squeeze(i) for i in lambdaHistory]
if retResidualNormsHistory:
residualNormsHistory = np.vsplit(residualNormsHistory,
np.shape(residualNormsHistory)[0])
residualNormsHistory = [np.squeeze(i) for i in residualNormsHistory]
if retLambdaHistory:
if retResidualNormsHistory:
return _lambda, blockVectorX, lambdaHistory, residualNormsHistory
else:
return _lambda, blockVectorX, lambdaHistory
else:
if retResidualNormsHistory:
return _lambda, blockVectorX, residualNormsHistory
else:
return _lambda, blockVectorX