We have introduced the generalized alternating direction implicit iteration (GADI) method for solving large sparse complex symmetric linear systems and proved its convergence properties. Additionally, some numerical results have demonstrated the effectiveness of this algorithm. Furthermore, as an application of the GADI method in solving complex symmetric linear systems, we utilized the flattening operator and Kronecker product properties to solve Lyapunov and Riccati equations with complex coefficients using the GADI method. In solving the Riccati equation, we combined inner and outer iterations, first simplifying the Riccati equation into a Lyapunov equation using the Newton method, and then applying the GADI method for solution. Finally, we provided convergence analysis of the method and corresponding numerical results.

翻译：我们提出了用于求解大型稀疏复杂对称线性系统的广义交替方向隐式迭代（GADI）方法，并证明了其收敛性。此外，数值结果验证了该算法的有效性。进一步地，作为GADI方法在复杂对称线性系统求解中的应用，我们利用展平算子和克罗内克积性质，将GADI方法应用于求解具有复系数的李雅普诺夫方程和里卡蒂方程。在求解里卡蒂方程时，我们结合了内外迭代策略，首先通过牛顿法将里卡蒂方程简化为李雅普诺夫方程，随后应用GADI方法进行求解。最后，我们给出了该方法的收敛性分析及相应的数值结果。