  For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new
general theory, independent of iterative solvers for shifted inner linear
systems. The theory shows that the method converges at least quadratically
under a new condition, called the uniform positiveness condition, that may
allow inner tolerance $\xi_k\geq 1$ at outer iteration $k$ and can be
considerably weaker than the condition $\xi_k\leq\xi<1$ with $\xi$ a constant
not near one commonly used in literature. We consider the convergence of the
inexact RQI with the unpreconditioned and tuned preconditioned MINRES method
for the linear systems. Some attractive properties are derived for the
residuals obtained by MINRES. Based on them and the new general theory, we make
a more refined analysis and establish a number of new convergence results. Let
$\|r_k\|$ be the residual norm of approximating eigenpair at outer iteration
$k$. Then all the available cubic and quadratic convergence results require
$\xi_k=O(\|r_k\|)$ and $\xi_k\leq\xi$ with a fixed $\xi$ not near one,
respectively. Fundamentally different from these, we prove that the inexact RQI
with MINRES generally converges cubically, quadratically and linearly provided
that $\xi_k\leq\xi$ with a constant $\xi<1$ not near one, $\xi_k=1-O(\|r_k\|)$
and $\xi_k=1-O(\|r_k\|^2)$, respectively. Therefore, the new convergence
conditions are much more relaxed than ever before. The theory can be used to
design practical stopping criteria to implement the method more effectively.
Numerical experiments confirm our results.
