This paper proposes an algorithm to compute solutions X to the linear matrix equation and inequality of the type(I - BB+)(AX + XA＇ + W)(I - BB+) = 0, X > 0.This problem arises in the synthesis of covariance controllers; the set of symmetric matrices X assignable as a closed-loop state covariance by a stabilizing controller is characterized by these conditions. Our algorithm generates analytical solutions to the above problem in a recursive manner. In this sense, our algorithm is essentially different from other computational methods pertinent to this problem, such as convex programming. As a result, the algorithm does not involve the issue of convergence and terminates in an a priori known finite number of steps. Thus, the computational complexity is expected to be much less than that of other methods.