On the correction equation of the Jacobi-Davidson method

TL;DR

This paper analyzes the correction equation in the Jacobi-Davidson method, revealing conditions for its solvability and linking stagnation to defective eigenvalues, thus improving understanding of the method's convergence.

Contribution

It provides necessary and sufficient conditions for the existence and uniqueness of solutions to the correction equation, and clarifies the relationship between stagnation and defective eigenvalues.

Findings

Correction equation may have no solution or a unique solution.

Stagnation occurs when the Ritz value is a defective eigenvalue.

Conditions for successful subspace expansion are established.

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 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.