A numerical algorithm to solve AT X A - X = Q

Two kinds of algorithm are usually resorted to in order to solve the well-known Lyapounov discrete equation AT X A - X = Q : transformation of the original linear system in a classical one with n(n+1)/2 unknowns, and iterative scheme [1]. The first requires n4/4 storage words and a cost of n6/3 multiplications, which is impractical with a large system, and the second applies only if A is a stable matrix. The solution proposed requires no stability assumption and operates in only some n2 words and n3 multiplications.