Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method 论文

摘要

. 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 dierent methods in the class, with dierent 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 nding 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 ecient as the \\i...