Numerical methods for the QCD overlap operator: III. Nested iterations

In this paper we investigate aspects of this nested paradigm. We examine several Krylov subspace methods to be used as an outer iteration for both propagator computations and the Hybrid Monte-Carlo scheme. We establish criteria on the accuracy of the inner iteration which allow to preserve an a priori given precision for the overall computation. It will turn out that the accuracy of the sign function can be relaxed as the outer iteration proceeds. Furthermore, we consider preconditioning strategies, where the preconditioner is built upon an inaccurate approximation to the sign function. Relaxation combined with preconditioning allows for considerable savings in computational efforts up to a factor of 4 as our numerical experiments illustrate. We also discuss the possibility of projecting the squared overlap operator into one chiral sector.