  In this article, we consider solvers for large-scale trust-region subproblems
when the quadratic model is defined by a limited-memory symmetric rank-one
(L-SR1) quasi-Newton matrix. We propose a solver that exploits the compact
representation of L-SR1 matrices. Our approach makes use of both an orthonormal
basis for the eigenspace of the L-SR1 matrix and the Sherman-Morrison-Woodbury
formula to compute global solutions to trust-region subproblems. To compute the
optimal Lagrange multiplier for the trust-region constraint, we use Newton's
method with a judicious initial guess that does not require safeguarding. A
crucial property of this solver is that it is able to compute high-accuracy
solutions even in the so-called hard case. Additionally, the optimal solution
is determined directly by formula, not iteratively. Numerical experiments
demonstrate the effectiveness of this solver.
