  The main focus in this paper is exact linesearch methods for minimizing a
quadratic function whose Hessian is positive definite. We give a class of
limited-memory quasi-Newton Hessian approximations which generate search
directions parallel to those of the method of preconditioned conjugate
gradients, and hence give finite termination on quadratic optimization problems
in exact arithmetic. With the framework of reduced-Hessians this class provides
a dynamical framework for the construction of limited-memory quasi-Newton
methods. We give an indication of the performance of the methods within this
framework by showing numerical simulations on sequences of related systems of
linear equations, which originate from the CUTEst test collection. In addition,
we give a compact representation of the Hessian approximations in the full
Broyden class for the general unconstrained optimization problem. This
representation consists of explicit matrices and gradients only as vector
components.
