Inverse Iteration is a well known and well studied method for the calculation of A x =λ x, which requires solves with A-σ I where σ is the shift. However for large sparse matrices one is usually forced to use iterative methods for the linear systems and so (A-σ I)y=x is satisfied to a residual tolerance τ. Due to this one obtains an inner-outer method with two parameters σ and τ. Questions of convergence and optimal choices for the parameters need to be answered. In previous contributions [1] and [2] we provided a general convergence result. Further we showed that it is beneficial to use the Rayleigh quotient as shift when Galerkin Krylov solvers are applied to the linear systems. Recently [3] showed that it is important to change the right hand side of the linear systems when Cholesky preconditioning is used. In [1] and [2] we extend this result to arbitrary preconditioners.
This talk will outline theoretical treatment, present numerical result based on examples from matrix market, including a comparison with LOBPCG, and discuss in detail implementational issues.

[1] Berns-Mueller, Joerg: Inexact Inverse Iteration using Galerkin Krylov solvers, PhD thesis, University of Bath. to be submitted.
[2] Berns-Mueller, J., I. G. Graham, A. Spence: Inverse Iteration and Inexact Solves. to appear.
[3] Simoncini, Valeria and Elden, Lars (2002): Inexact Rayleigh quotient-type methods for eigenvalue computations, BIT 42(1), 159-182