  Trust-region subproblem (TRS) is an important problem arising in many
applications such as numerical optimization, Tikhonov regularization of
ill-posed problems, and constrained eigenvalue problems. In recent decades,
extensive works focus on how to solve the trust-region subproblem efficiently.
To the best of our knowledge, there are few results on perturbation analysis of
the trust-region subproblem. In order to fill in this gap, we focus on
first-order perturbation theory of the trust-region subproblem. The main
contributions of this paper are three-fold. First, suppose that the TRS is in
\emph{easy case}, we give a sufficient condition under which the perturbed TRS
is still in easy case. Second, with the help of the structure of the TRS and
the classical eigenproblem perturbation theory, we perform first-order
perturbation analysis on the Lagrange multiplier and the solution of the TRS,
and define their condition numbers. Third, we point out that the solution and
the Lagrange multiplier could be well-conditioned even if TRS is in {\it nearly
hard case}. The established results are computable, and are helpful to evaluate
ill-conditioning of the TRS problem beforehand. Numerical experiments show the
sharpness of the established bounds and the effectiveness of the proposed
strategies.
