EFFICIENT MATRIX-VALUED ALGORITHMS FOR SOLVING STIFF RICCATI DIFFERENTIAL-EQUATIONS

One of the most deeply studied nonlinear matrix differential equations arising in science and engineering, particularly in optimal control and filtering, is the Riccati differential equation (RDE). In the time-varying case, a classical approach which has been widely used to compute the solution of the matrix equation of size say, n × n, is to unroll the matrices into vectors and integrate the resulting system of n2 vector differential equations directly. If the system of vectorized differential equations is stiff, the cost (computation time and storage requirements) of applying the popular backward differentiation formulas (BDF's) to the stiff equations will be very high for large n because a linear system of algebraic equations of size n2 xn2 must be solved at each time step. In this paper, new matrix-valued algorithms based on a matrix generalization of the BDF's are proposed for solving stiff RDE's. The amount of work required to compute the solution per time step is only 0(n3) flops by using the matrix-valued algorithms, whereas the classical approach requires 0(n6) flops per time step. © 1990 IEEE