On the correction equation of the Jacobi–Davidson method

Abstract

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.