NUMERICAL-INTEGRATION OF THE DIFFERENTIAL RICCATI EQUATION AND SOME RELATED ISSUES

In this paper the problem of direct numerical integration of differential Riccati equations (DREs) and some related issues are considered. The DRE is an expression of a particular change of variables for a linear system of ordinary differential equations. The error that an approximate solution of the DRE induces on the original variables of the system is considered, and it is related to geometrical properties of the system itself. Sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system. Nonstiff and stiff DREs of unsymmetric and symmetric type are considered. A useful matrix interpretation is given for many integration schemes (such as the backward differentiation formulas, BDF), when applied to the DRE. This allows the matrix structure of the problem to be exploited. In particular, for stiff DREs, the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reformulation of the DRE as a system of vector equations. Structure preserving properties of several schemes are shown for symmetric DREs. Furthermore, it is shown how the BDF allow one to obtain important eigenvalue information. A FORTRAN code, DRESOL, is presented, which is designed for the direct numerical integration of DREs. Implementation considerations for this code and a few examples to show its performance are provided. The code is built around the well known solver LSODE, but, although it shares much of LSODE philosophy, DRESOL presents a number of original features which make it more efficient and reliable than LSODE for the integration of DREs. For example, DRESOL has a completely redesigned linear algebra module, it uses matrix arithmetic throughout to enhance efficiency, and it handles automatically the symmetric or nonsymmetric cases. Also, DRESOL automatically provides eigenvalue information, which has been used, along with the standard local error estimates, to perform a control on the step-size selection, thus successfully avoiding the practical occurrences of superstability for the BDF.