A Natural Approach to the Numerical Integration of Riccati Differential Equations

This paper introduces a new class of methods, which we call Möbius schemes, for the numerical solution of matrix Riccati differential equations. The approach is based on viewing the Riccati equation in its natural geometric setting, as a flow on the Grassmanian of m-dimensional subspaces of an (n + m)-dimensional vector space. Since the Grassmanians are compact differentiable manifolds, and the coefficients of the equation are assumed continuous, there are no singularities or intrinsic instabilities in the associated flow. The presence of singularities and numerical instabilitites is an artefact of the coordinate system, but since Möbius schemes are based on the natural geometry, they are able to deal with numerical instability and pass accurately through the singularities. A number of examples are given to demonstrate these properties.