  We investigate fast direct methods for solving systems of the form (B + G)x =
y, where B is a limited-memory BFGS matrix and G is a symmetric
positive-definite matrix. These systems, which we refer to as shifted L-BFGS
systems, arise in several settings, including trust-region methods and
preconditioning techniques for interior-point methods. We show that under mild
assumptions, the system (B + G)x = y can be solved in an efficient and stable
manner via a recursion that requies only vector inner products. We consider
various shift matrices G and demonstrate the effectiveness of the recursion
methods in numerical experiments.

Numerical Analysis