Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods

The Davidson method is a popular preconditioned variant of the Arnoldi method for solving large eigenvalue problems. For theoretical as well as practical reasons the two methods are often used with restarting. Frequently, information is saved through approximated eigenvectors to compensate for the convergence impairment caused by restarting. We call this scheme of retaining more eigenvectors than needed "thick restarting" and prove that thick restarted, nonpreconditioned Davidson is equivalent to the implicitly restarted Arnoldi. We also establish a relation between thick restarted Davidson and a Davidson method applied on a deflated system. The theory is used to address the question of which and how many eigenvectors to retain and motivates the development of a dynamic thick restarting scheme for the symmetric case, which can be used in both Davidson and implicit restarted Arnoldi. Several experiments demonstrate the efficiency and robustness of the scheme.