Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration

Convergence results are provided for inexact two-sided inverse and Rayleigh quotient iteration, which extend the previously established results to the generalized non-Hermitian eigenproblem and inexact solves with a decreasing solve tolerance. Moreover, the simultaneous solution of the forward and adjoint problem arising in two-sided methods is considered, and the successful tuning strategy for preconditioners is extended to two-sided methods, creating a novel way of preconditioning two-sided algorithms. Furthermore, it is shown that inexact two-sided Rayleigh quotient iteration and the inexact two-sided Jacobi-Davidson method (without subspace expansion) applied to the generalized preconditioned eigenvalue problem are equivalent when a certain number of steps of a Petrov-Galerkin-Krylov method is used and when this specific tuning strategy is applied.