Numerical Integration Of Matrix Riccati Differential Equations With Solution Singularities

Abstract

A matrix Riccati differential equation (MRDE) is a quadratic ODE of the form X' = A₂₁ + A₂₂X - XA₁₁ - XA₁₂X ; where X is a function of t with X : R Rnxm and the Aij's are constant or functions of t with matrix sizes to respect the size of X. It is well known that MRDEs may have singularities in their solution even if all the Aij are constant. In this dissertation, several di erent ideas for the meaning of the solution of an MRDE past a solution singularity are analyzed and it is shown how all these ideas are related. Then, a class of numerical methods are given which respect all these ideas. Finally, a robust numerical integration scheme is given based on these numerical methods and several examples are shown to validate the numerical integration scheme.