engleski

The invariant subspace approach to solving the Riccati equation

We consider an approach to solving a large-scale algebraic Riccati equation A'X + XA + C'C - X BB' X=0 via computing a small-dimensional stable invariant subspace of the associated Hamiltonian matrix H. Properties of the Riccati equation that imply the existence of a low-rank stabilizing solution are discussed, and we address two issues in order to obtain a good approximation to such a solution: how to determine which eigenpairs of H are to be computed, and how to construct the solution once the invariant subspace is obtained. Commonly used projection methods for the Riccati equation are usually based on approximations from the Krylov or rational Krylov subspaces generated by the matrices A' and the inverse of A', and the initial (block-)vector C'. We also give some new insights into these methods by drawing connections between them and the symplectic Lanczos processes for the Hamiltonian matrix.

matrix equations; algebraic Riccati equations; Hamiltonian matrices; invariant subspaces; Krylov subspaces; ADI iteration

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