Square-root algorithms for the continuous-time linear least-square estimation problem

We present a simple differential equation for the triangular square root of the state error variance of the continuous-time Kalman filter. Unlike earlier methods of Andrews, and Tapley and Choe, this algorithm does not explicitly involve any antisymmetric matrix in the differential equation for the square roots. The role of antisymmetric matrices is clarified: it is shown that they are just the generators of the orthogonal transformations that connect the various square roots; in the constant model case, a similar set of antisymmetric matrices appears inside the Chandrasekhar-type equations for the square roots of the derivative of the error variance. Several square-root algorithms for the smoothing problem are also presented and are related to some well-known smoothing approaches.