Efficient matrix-valued algorithms for solving stiff Riccati differential equations

In the time-varying case, a classical approach which has been widely used to compute the solution of the Riccati matrix equation of, for example, size 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 (BDFs) to the stiff equations will be very high for large n because a linear system of algebraic equations of size n²×n² must be solved at each time step. New matrix-valued algorithms based on a matrix generalization of the BDFs are proposed for solving stiff Riccatti differential equations. The amount of work required to compute the solution per time step is only O(n³) flops by using the matrix-valued algorithms, whereas the classical approach requires O(n⁶) flops per time step