We describe the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) Method for symmetric eigenvalue problems, based on a local optimization of a three-term recurrence. We suggest using the same multigrid preconditioner in the LOBPCG method for eigenproblems as that in the preconditioned conjugate gradient method for the corresponding system of linear equations. We provide new convergence rate estimates and numerical results, which show effectiveness of such an approach.

A MATLAB code of the LOBPCG method is available at http://www-math.cudenver.edu/~aknyazev/software/CG/

The talk is partially based on the papers:

• Andrew Knyazev, Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method, Report UCD-CCM 149, 2000, at the Center for Computational Mathematics, University of Colorado at Denver (a revised version accepted to SIAM SISC).
• Klaus Neymeyr, Solving mesh eigenproblems with multigrid efficiency, SFB 382, Report Nr. 157, October 2000.