On convergence of iterative projection methods for symmetric eigenvalue problems

We prove global convergence of particular iterative projection methods using the so-called shift-and-invert technique for solving symmetric generalized eigenvalue problems. In particular, we aim to provide a variant of the convergence theorem obtained by Crouzeix, Philippe, and Sadkane for the generalized Davidson method. Our result covers the Jacobi–Davidson and the rational Krylov methods with restarting and preconditioning that are important techniques for modern eigensolvers. More specifically, we prove that the Ritz pairs converge to exact eigenpairs, even though they are not necessarily the target eigenpairs. We would like to emphasize that our proof is not a routine consideration of Crouzeix, Philippe, and Sadkane. To complete the proof, we discover a key lemma, which leads to a very simple convergence proof, resulting in a new theorem similar to that of Crouzeix, Philippe, and Sadkane.