Inexact Newton’s method with inner implicit preconditioning for algebraic Riccati equations

Abstract

Continuous algebraic Riccati equations (CARE) appear in several important applications. A suitable approach for solving CARE, in the large-scale case, is to apply Kleinman–Newton’s method which involves the solution of a Lyapunov equation at every inner iteration. Lyapunov equations are linear, nevertheless, solving them requires specialized techniques. Different numerical methods have been designed to solve them, including ADI and Krylov-type iterative projection methods. For these iterative schemes, preconditioning is always a difficult task that can significantly accelerate the convergence. We present and analyze a strategy for solving CARE based on the use of inexact Kleinman–Newton iterations with an implicit preconditioning strategy for solving the Lyapunov equations at each inner step. One advantage is that the Newton direction is approximated implicitly, avoiding the explicit knowledge of the given matrices. Only the effect of the matrix–matrix products with the given matrices is required. We present illustrative numerical experiments on some test problems.