Symplectic Methods for the Solution to Riccati Matrix Equations Related to Macroeconomic Models

This paper deals with the numerical solution of the differential Riccati equation associated with deterministic time invariant linear-quadratic control problems. We focus on the solution of the differential Riccati equations via a hamiltonian equation, using symplectic numerical methods. These methods are appropriate for long time integration and preserve some of the important qualitative features of the problem, such as the symmetry and positivity of the Riccati matrix. The numerical methods can be also interpreted as iterative methods for the algebraic equation. We present some numerical results and a first analytical advance. A macroeconomic linear-quadratic control problem is used as a test problem in the numerical experiments.