We present a new method for the large-scale trust-region subproblem. The me
thod is matrix-free in the sense that only matrix-vector products are requi
red. We recast the trust-region subproblem as a parameterized eigenvalue pr
oblem and compute an optimal value for the parameter. We then nd the soluti
on of the trust-region subproblem from the eigenvectors associated with two
of the smallest eigenvalues of the parameterized eigenvalue problem corres
ponding to the optimal parameter. The new algorithm uses a different interp
olating scheme than existing methods and introduces a uni ed iteration that
naturally includes the so-called hard case. We show that the new iteration
is well defined and convergent at a superlinear rate. We present computati
onal results to illustrate convergence properties and robustness of the met
hod.