The projected Newton-Kleinman method for the algebraic Riccati equation
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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 that guarantees the convergence of the inexact procedure to a stabilizing solution turns out to be exact in our setting. 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.
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