  The numerical solution of the algebraic Riccati equation is a challenging
task especially for very large problem dimensions. In this paper we present a
new algorithm that combines the very appealing computational features of
projection methods with the convergence properties of the inexact
Newton-Kleinman procedure equipped with a line search. In particular, the
Newton scheme is completely merged in a projection framework with a single
approximation space so that the Newton-Kleinman iteration is only implicitly
performed. Moreover, the line search turns out to be exact in our setting,
i.e., the existence of a local minimum of the Riccati residual norm along the
current search direction is guaranteed and the corresponding minimizer is
chosen as step-size. This property determines a monotone decrease of the
Riccati residual norm under some mild assumptions. Several numerical results
are reported to illustrate the potential of our novel approach.
