A numerical algorithm to solveA^{T}XA - X = Q

Two kinds of algorithms are usually resorted to in order to solve the well-known Lyapounov discrete equation
A^{T}XA - 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
n^{4}/4
storage words and a cost of
n^{6}/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 n
²
words and n
³
multiplications.