Generalized Chandrasekhar algorithms: Time-varying models

The Riccati equation (RE) plays a fundamental role in optimal control theory, linear estimation, radiative transfer, neutron transport theory, etc. Its effective, numerical solution constitutes the integral prerequisite to the solution of important problems in the above and related fields. A computationally advantageous approach to the solution of matrix Re's is the so-called
x-y
or Chandrasekhar algorithm through which the matrix RE is replaced by two coupled differential equations of lesser dimensionality. These previous Chandrasekhar algorithms were, however, restricted to the case of time-invariant models. In this short paper, generalized
x-y
or Chandrasekhar algorithms are presented that are applicable to time-varying models as well as time-invariant ones. Backward and forward time differentiations are introduced that readily yield the generalized Chandrasekhar algorithms as well as provide several interesting interpretations of these results. Furthermore, the possible computational advantages, as well as the theoretical significance of the generalized Chandrasekhar algorithms are explored.