Abstract
The generalized Davidson (GD) method can be viewed as a generalization of the preconditioned steepest descent (PSD) method for solving symmetric eigenvalue problems. In the GD method, the new approximation is sought in the subspace that spans all the previous approximate eigenvectors, in addition to the current one and the preconditioned residual thereof used in PSD. In this respect, the relation between the GD and PSD methods is similar to that between the standard steepest descent method for linear systems and methods in Krylov subspaces. This paper presents convergence estimates for the (restarted) GD method that demonstrate convergence acceleration compared to the PSD method, similar to that achieved by methods in Krylov subspaces compared to the standard steepest descent.
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