Locking issues for finding a large number of eigenvectors of Hermitian matrices ?

Locking is a popular deflation technique followed by many eigensolvers, where a converged eigenvector is frozen and removed from the iteration search space. Other deflation techniques that do not modify the matrix have less favorable numerical properties, so the alternative to locking is not to perform deflation at all. Without deflation, which we refer to as non-locking, converged eigenvectors are kept in the search space. One of the goals of this paper is to determine when locking is computationally preferable, and for which eigensolvers. Our primary goal, however, is to address a subtle numerical, but not floating point, problem that arises with locking. The problem stems from the fact that converged eigenpairs are only accurate to a specified tolerance Tol, so if they are locked, they may impede convergence to Tol accuracy for some subsequent eigenvector. Although the problem is rare, the resulting stagnation is a liability for general purpose software. We provide a theoretical explanation of the problem, and based on it we derive an algorithm that resolves the problem, is easy to implement, and incurs minimal additional costs.