Pure SciPy implementation of Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG), see

License: BSD

Authors: Robert Cimrman, Andrew Knyazev

Examples in tests directory contributed by Nils Wagner.

Module Contents


as2d(ar) If the input array is 2D return it, if it is 1D, append a dimension,
_makeOperator(operatorInput,expectedShape) Takes a dense numpy array or a sparse matrix or
_applyConstraints(blockVectorV,factYBY,blockVectorBY,blockVectorY) Changes blockVectorV in place.
lobpcg(A,X,B=None,M=None,Y=None,tol=None,maxiter=20,largest=True,verbosityLevel=0,retLambdaHistory=False,retResidualNormsHistory=False) Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG)
save(ar, fileName)
_assert_symmetric(M, rtol=1e-05, atol=1e-08)

If the input array is 2D return it, if it is 1D, append a dimension, making it a column vector.

_makeOperator(operatorInput, expectedShape)

Takes a dense numpy array or a sparse matrix or a function and makes an operator performing matrix * blockvector products.

>>> A = _makeOperator( arrayA, (n, n) )
>>> vectorB = A( vectorX )
_applyConstraints(blockVectorV, factYBY, blockVectorBY, blockVectorY)

Changes blockVectorV in place.

_b_orthonormalize(B, blockVectorV, blockVectorBV=None, retInvR=False)
lobpcg(A, X, B=None, M=None, Y=None, tol=None, maxiter=20, largest=True, verbosityLevel=0, retLambdaHistory=False, retResidualNormsHistory=False)

Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG)

LOBPCG is a preconditioned eigensolver for large symmetric positive definite (SPD) generalized eigenproblems.

A : {sparse matrix, dense matrix, LinearOperator}
The symmetric linear operator of the problem, usually a sparse matrix. Often called the “stiffness matrix”.
X : array_like
Initial approximation to the k eigenvectors. If A has shape=(n,n) then X should have shape shape=(n,k).
B : {dense matrix, sparse matrix, LinearOperator}, optional
the right hand side operator in a generalized eigenproblem. by default, B = Identity often called the “mass matrix”
M : {dense matrix, sparse matrix, LinearOperator}, optional
preconditioner to A; by default M = Identity M should approximate the inverse of A
Y : array_like, optional
n-by-sizeY matrix of constraints, sizeY < n The iterations will be performed in the B-orthogonal complement of the column-space of Y. Y must be full rank.
w : array
Array of k eigenvalues
v : array
An array of k eigenvectors. V has the same shape as X.
tol : scalar, optional
Solver tolerance (stopping criterion) by default: tol=n*sqrt(eps)
maxiter : integer, optional
maximum number of iterations by default: maxiter=min(n,20)
largest : bool, optional
when True, solve for the largest eigenvalues, otherwise the smallest
verbosityLevel : integer, optional
controls solver output. default: verbosityLevel = 0.
retLambdaHistory : boolean, optional
whether to return eigenvalue history
retResidualNormsHistory : boolean, optional
whether to return history of residual norms

Solve A x = lambda B x with constraints and preconditioning.

>>> from scipy.sparse import spdiags, issparse
>>> from scipy.sparse.linalg import lobpcg, LinearOperator
>>> n = 100
>>> vals = [np.arange(n, dtype=np.float64) + 1]
>>> A = spdiags(vals, 0, n, n)
>>> 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.]])


>>> Y = np.eye(n, 3)

Initial guess for eigenvectors, should have linearly independent columns. Column dimension = number of requested eigenvalues.

>>> X = np.random.rand(n, 3)

Preconditioner – inverse of A (as an abstract linear operator).

>>> invA = spdiags([1./vals[0]], 0, n, n)
>>> def precond( x ):
...     return invA  * x
>>> M = LinearOperator(matvec=precond, shape=(n, n), dtype=float)

Here, invA could of course have been used directly as a preconditioner. Let us then solve the problem:

>>> eigs, vecs = lobpcg(A, X, Y=Y, M=M, tol=1e-4, maxiter=40, largest=False)
>>> eigs
array([ 4.,  5.,  6.])

Note that the vectors passed in Y are the eigenvectors of the 3 smallest eigenvalues. The results returned are orthogonal to those.

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 m the number of required eigenvalues (smallest or largest).

The LOBPCG code internally solves eigenproblems of the size 3``m`` on every iteration by calling the “standard” dense eigensolver, so if m is not small enough compared to n, it does not make sense to call the LOBPCG code, but rather one should use the “standard” eigensolver, e.g. numpy or scipy function in this case. If one calls the LOBPCG algorithm for 5``m``>``n``, it will most likely break internally, so the code tries to call the standard function instead.

It is not that n should be large for the LOBPCG to work, but rather the ratio n/m should be large. It you call the LOBPCG code with m``=1 and ``n``=10, it should work, though ``n is small. The method is intended for extremely large n/m, see e.g., reference [28] in

The convergence speed depends basically on two factors:

  1. How well relatively separated the seeking eigenvalues are from the rest of the eigenvalues. One can try to vary m to make this better.
  2. How well conditioned the problem is. This can be changed by using proper preconditioning. 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.

Acknowledgements code was written by Robert Cimrman. Many thanks belong to Andrew Knyazev, the author of the algorithm, for lots of advice and support.

[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.
[2]A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov (2007), Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in hypre and PETSc.
[3]A. V. Knyazev’s C and MATLAB implementations: