A. V. Knyazev
January, 2000
URN = ncstrl.cudenver_ccm/UCD-CCM-149
URL =
http://cs-tr.cs.cornell.edu:80/Dienst/UI/1.0/Display/ncstrl.cudenver_ccm/UCD-CCM-149
Abstract: We describe new algorithms of the Locally Optimal Block
Preconditioned Conjugate Gradient (LOBPCG) Method for symmetric eigenvalue
problems, based on a local optimization of a three-term recurrence. To be
able to compare numerically different methods in the class, with different
preconditioners, we suggest a common system of model tests, using random
preconditioners and initial guesses. As the ``ideal'' control algorithm, we
propose the standard preconditioned conjugate gradient method for finding an
eigenvector as an element of the null--space of the corresponding homogeneous
system of linear equations under the assumption that the eigenvalue is known.
We recommend that every new preconditioned eigensolver be compared with this
``ideal'' algorithm on our model test problems in terms of the speed of
convergence, costs of every iterations and memory requirements. We provide
such comparison for our LOBPCG Method. Numerical results establish that our
algorithm is practically as efficient as the ``ideal'' algorithm when the same
preconditioner is used in both methods. We also show numerically that the
LOBPCG Method provides approximations to first eigenpairs of about the same
quality as those by the much more expensive global optimization method on the
same generalized block Krylov subspace. Finally, direct numerical
comparisons with the Jacobi--Davidson method show that our method is
more robust and converges almost two times faster.