lobpcg — SciPy v1.15.3 Manual (original) (raw)
scipy.sparse.linalg.
scipy.sparse.linalg.lobpcg(A, X, B=None, M=None, Y=None, tol=None, maxiter=None, largest=True, verbosityLevel=0, retLambdaHistory=False, retResidualNormsHistory=False, restartControl=20)[source]#
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”.
Xndarray, 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.
Yndarray, 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.
tolscalar, optional
The default is tol=n*sqrt(eps). Solver tolerance for the stopping criterion.
maxiterint, default: 20
Maximum number of iterations.
largestbool, default: True
When True, solve for the largest eigenvalues, otherwise the smallest.
verbosityLevelint, optional
By default verbosityLevel=0 no output. Controls the solver standard/screen output.
retLambdaHistorybool, default: False
Whether to return iterative eigenvalue history.
retResidualNormsHistorybool, default: False
Whether to return iterative history of residual norms.
restartControlint, 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:
lambdandarray of the shape (k, ).
Array of k approximate eigenvalues.
vndarray of the same shape as X.shape.
An array of k approximate eigenvectors.
lambdaHistoryndarray, optional.
The eigenvalue history, if retLambdaHistory is True.
ResidualNormsHistoryndarray, 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=1and 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:
- Quality of the initial approximations X to the seeking eigenvectors. Randomly distributed around the origin vectors work well if no better choice is known.
- Relative separation of the desired eigenvalues from the rest of the eigenvalues. One can vary
kto improve the separation. - 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
Examples
Our first example is minimalistic - find the largest eigenvalue of a diagonal matrix by solving the non-generalized eigenvalue problemA 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
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.float32forces 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 productA @ 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
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 shiftingvals 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.])