A natural approach to the numerical integration of Riccati differential equations

This paper introduces a new class of methods, which we call Mobius schemes, for the numerical solution of matrix Riccati differential equations. The a pproach is based on viewing the Riccati equation in its natural geometric s etting, as a flow on the Grassmannian of m-dimensional subspaces of an (n+m )-dimensional vector space. Since the Grassmannians are compact differentia ble manifolds, and the coefficients of the equation are assumed continuous, there are no singularities or intrinsic instabilities in the associated fl ow. The presence of singularities and numerical instabilities is an artifac t of the coordinate system, but since Mobius schemes are based on the natur al geometry, they are able to deal with numerical instability and pass accu rately through the singularities. A number of examples are given to demonst rate these properties.