An invariant subspace method for large-scale algebraic Riccati equation

This paper introduces a new family of low-rank approximations of the solution of the algebraic Riccati equation by considering stable invariant subspaces of the Hamiltonian matrix. They are defined from an appropriate expression of the solution of the ARE which generalizes to stable invariant subspaces of any dimension. The main features of the exact solution, in particular the positive semi-definiteness, are preserved. In the case of algebraic Bernoulli equation we obtain the exact solution and a direct method to compute it. Numerical examples illustrate the effectiveness of the proposed approach.