On the correction equation of the Jacobi-Davidson method
This work clarifies fundamental properties of the Jacobi-Davidson method for numerical linear algebra researchers, but the results are incremental as they refine existing understanding rather than introducing a new paradigm.
The authors analyze the Jacobi-Davidson correction equation, showing it may have a unique solution or no solution, and derive a computable condition for judging existence and uniqueness. They also link stagnation to defective Ritz values and provide a condition for successful subspace expansion.
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 long by scholars that the Jacobi-Davidson correction equation is a consistent linear system. In this work, we point out that the correction equation may have a unique solution or have no solution at all, and we derive a computable necessary and sufficient condition for cheaply judging the existence and uniqueness of solution of the correction equation. Furthermore, we consider the difficulty of stagnation that bothers the Jacobi-Davidson method, and verify that if the Jacobi-Davidson method stagnates, then the corresponding Ritz value is a defective eigenvalue of the projection matrix. We provide a computable necessary and sufficient condition for expanding the search subspace successfully. The properties of the Jacobi-Davidson method with preconditioning and some alternative Jacobi-Davidson correction equations are also discussed.