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.

翻译：本文提出了一种新的分裂方法，用于求解形如$AP + PA^T=-BB^T$的Lyapunov方程，并将其表示为Kronecker积形式。尽管所得矩阵的阶数为$n^2$，但该方法每次迭代仅需对矩阵$A$进行两种运算：形如$(A-\sigma I) \hat{B}$的乘法运算和形如$(A-\sigma I)^{-1}\hat{B}$的求逆运算。我们证明，对于某些参数选择，迭代矩阵的所有特征值的绝对值均小于1，这意味着该方法无需依赖初始向量即可收敛。然而，我们通过一个定理说明了如何为该方法选取一个良好的初始向量。