Power series evaluation of transition and covariance matrices

Power series solutions to the matrix covariance differential equation
\dot{P} = AP + (AP)' + Q
and the transition differential equation
\dot{\Phi} = A\Phi
are reexamined. Truncation error bounds are derived which are computationally attractive and which extend previous results. Polynomial approximations are obtained by exploiting the functional equations satisfied by the transition and covariance matrices. The series-functional equation propagation technique represents a fast and accurate alternative to the numerical integration of the time-invariant transition and covariance equations.