Combined method for the solution of asymmetric Riccati differential equations

The combination of two methods is described for the solution of general matrix Riccati differential equation, namely the computer precise integration method and the analytical method based on eigensolutions. The two-point boundary value problem of a linear system is closely related to the m× n solution matrix of the Riccati differential equations. The corresponding boundary conditions at two ends, t= t 0 and t= t f, of the related linear system are m and n vectors, respectively. According to both the 2 N algorithm and the ill-condition avoiding technique, the computer precise integration method is proposed first, which is always numerically stable. Then based on eigensolutions of the related linear system, the analytical solutions of Riccati differential equations both for backward and for forward integration versions, are proposed, which is quite efficient but has a numerical problem when the Jordan normal form is nearly to appear. The combination of both methods gives good results. Numerical examples show that for the m× n matrix solution, the numerical results obtained from the present analytical method coincide with the precise integration method up to 10 digits.