  For the Hermitian inexact Rayleigh quotient iteration (RQI), the author has
established new local general convergence results, independent of iterative
solvers for inner linear systems. The theory shows that the method locally
converges quadratically under a new condition, called the uniform positiveness
condition. In this paper we first consider the local convergence of the inexact
RQI with the unpreconditioned Lanczos method for the linear systems. Some
attractive properties are derived for the residuals, whose norms are
$\xi_{k+1}$'s, of the linear systems obtained by the Lanczos method. Based on
them and the new general convergence results, we make a refined analysis and
establish new local convergence results. It is proved that the inexact RQI with
Lanczos converges quadratically provided that $\xi_{k+1}\leq\xi$ with a
constant $\xi\geq 1$. The method is guaranteed to converge linearly provided
that $\xi_{k+1}$ is bounded by a small multiple of the reciprocal of the
residual norm $\|r_k\|$ of the current approximate eigenpair. The results are
fundamentally different from the existing convergence results that always
require $\xi_{k+1}<1$, and they have a strong impact on effective
implementations of the method. We extend the new theory to the inexact RQI with
a tuned preconditioned Lanczos for the linear systems. Based on the new theory,
we can design practical criteria to control $\xi_{k+1}$ to achieve quadratic
convergence and implement the method more effectively than ever before.
Numerical experiments confirm our theory.
