  We present here a new splitting method to solve Lyapunov equations of the
type $AP + PA^T=-BB^T$ in a Kronecker product form. Although that resulting
matrix is of order $n^2$, each iteration of the method demands only two
operations with the matrix $A$: a multiplication of the form $(A-\sigma I)
\hat{B}$ and a inversion of the form $(A-\sigma I)^{-1}\hat{B}$. We see that
for some choice of a parameter the iteration matrix is such that all their
eigenvalues are in absolute value less than 1, which means that it should
converge without depending on the starting vector. Nevertheless we present a
theorem that enables us how to get a good starting vector for the method.

Numerical Analysis;Numerical Analysis