  Solving the trust-region subproblem (TRS) plays a key role in numerical
optimization and many other applications. Based on a fundamental result that
the solution of TRS of size $n$ is mathematically equivalent to finding the
rightmost eigenpair of a certain matrix pair of size $2n$, eigenvalue-based
methods are promising due to their simplicity. For $n$ large, the implicitly
restarted Arnoldi (IRA) and refined Arnoldi (IRRA) algorithms are well suited
for this eigenproblem. For a reasonable comparison of overall efficiency of the
algorithms for solving TRS directly and eigenvalue-based algorithms, a vital
premise is that the two kinds of algorithms must compute the approximate
solutions of TRS with (almost) the same accuracy, but such premise has been
ignored in the literature. To this end, we establish close relationships
between the two kinds of residual norms, so that, given a stopping tolerance
for IRA and IRRA, we are able to determine a reliable one that GLTR should use
so as to ensure that GLTR and IRA, IRRA deliver the converged approximate
solutions with similar accuracy. We also make a convergence analysis on the
residual norms by the Generalized Lanczos Trust-Region (GLTR) algorithm for
solving TRS directly, the Arnoldi method and the refined Arnoldi method for the
equivalent eigenproblem. A number of numerical experiments are reported to
illustrate that IRA and IRRA are competitive with GLTR and IRRA outperforms
IRA.
