For the Hermitian inexact Rayleigh quotient iteration (RQI), the author has
established new general convergence results, independent of iterative solvers
for inner linear systems, indicating that the method converges quadratically
under a new condition, called the uniform positiveness condition. This
condition may allow inner tolerance $\xi_k\geq 1$ at outer iteration $k$ and is
weaker than the common $\xi_k<1$ not near one for quadratic convergence. In
this paper we consider the convergence of the inexact RQI with the Lanczos
method for the linear systems. Some attractive properties are derived for the
residuals obtained by Lanczos. Based on them and the new general convergence
results, we make a more refined analysis and establish new convergence results.
It is proved that the inexact RQI with Lanczos converges quadratically provided
that $\xi_k\leq\xi$ with a constant $\xi>1$, that is, the linear systems are
allowed to be solved with no accuracy in the sense of solving the linear
systems. The results are fundamentally different from the existing quadratic
convergence results and have an impact on implementations of the method. Based
on the new theory, we can design practical criteria to best control $\xi_k$ to
achieve quadratic convergence and implement the method more effectively than
ever before. Numerical experiments support our theory. The theory is extended
to the inexact RQI with a tuned preconditioned Lanczos for the linear systems.