  The generalized Lanczos trust-region (GLTR) method is one of the most popular
approaches for solving large-scale trust-region subproblem (TRS). Recently, Jia
and Wang [Z. Jia and F. Wang, \emph{SIAM J. Optim., 31 (2021), pp. 887--914}]
considered the convergence of this method and established some {\it a prior}
error bounds on the residual, the solution and the Largrange multiplier. In
this paper, we revisit the convergence of the GLTR method and try to improve
these bounds. First, we establish a sharper upper bound on the residual.
Second, we give a new bound on the distance between the approximation and the
exact solution, and show that the convergence of the approximation has nothing
to do with the associated spectral separation. Third, we present some
non-asymptotic bounds for the convergence of the Largrange multiplier, and
define a factor that plays an important role on the convergence of the
Largrange multiplier. Numerical experiments demonstrate the effectiveness of
our theoretical results.
