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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]], shape=(100, 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.], 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.], dtype=float32)
>>> eigenvalues, _ = lobpcg(A_matmat, X, maxiter=60)
>>> eigenvalues
array([100.], dtype=float32)
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.], dtype=float32)
The least efficient callable option is `aslinearoperator`:
>>> eigenvalues, _ = lobpcg(aslinearoperator(A), X, maxiter=80)
>>> eigenvalues
array([100.], dtype=float32)
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=90)
>>> 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.])
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