Positivity in numerical

In this work we address the issue of integrating symmetric Riccati and Lyapunov matrix diierential equations. In many cases { typical in applications { the solutions are positive deenite matrices. Our goal is to study when and how this property is maintained for a numerically computed solution. There are two classes of solution methods: direct and indirect algorithms. The rst class consists of the schemes resulting from direct discretization of the equations. The second class consists of algorithms which recover the solution by exploiting some special formulae that these solutions are known to satisfy. We show rst that using a direct algorithm { a one-step scheme or a strictly stable multistep scheme (explicit or implicit) { limits the order of the numerical method to one if we want to guarantee that the computed solution stays positive deenite. Then we show two ways to obtain positive deenite higher order approximations by using indirect algorithms. The rst is to apply a symplectic integrator to an associated Hamiltonian system. The other uses stepwise linearization.