Rayleigh quotient iteration and simplified Jacobi–Davidson method with preconditioned iterative solves

We show that for the non-Hermitian eigenvalue problem simplified Jacobi–Davidson with preconditioned iterative solves is equivalent to inexact Rayleigh quotient iteration where the preconditioner is altered by a simple rank one change. This extends existing equivalence results to the case of preconditioned iterative solves. Numerical experiments are shown to agree with the theory.