On the correction equation of the Jacobi-Davidson method

The Jacobi-Davidson method is one of the most popular approaches for iteratively computing a few eigenvalues and their associated eigenvectors of a large matrix. The key of this method is to expand the search subspace via solving the Jacobi-Davidson correction equation, whose coefficient matrix is singular. It is believed by scholars that the Jacobi-Davidson correction equation is consistent and has a unique solution. In this paper, however, we point out that the correction equation either has a unique solution or has no solution, and we derive a computable necessary and sufficient condition for cheaply judging the existence and uniqueness of the solution. Furthermore, we consider the problem of stagnation and verify that if the Jacobi-Davidson method stagnates, then the corresponding Ritz value is a defective eigenvalue of the projection matrix. Finally, we provide a computable criterion for expanding the search subspace successfully. The properties of some alternative Jacobi-Davidson correction equations are also discussed.