We derive the Alternating-Direction Implicit (ADI) method based on a commuting operator split and apply the results to the continuous time algebraic Lyapunov equation with low-rank constant term and approximate solution. Previously, it has been mandatory to start the low-rank ADI (LR-ADI) with an all-zero initial value. Our approach retains the known efficient iteration schemes of low-rank increments and residual to arbitrary low-rank initial values for the LR-ADI method. We further generalize some of the known properties of the LR-ADI for Lyapunov equations to larger classes of algorithms or problems. We investigate the performance of arbitrary initial values using two outer iterations in which LR-ADI is typically called. First, we solve an algebraic Riccati equation with the Newton method. Second, we solve a differential Riccati equation with a first-order Rosenbrock method. Numerical experiments confirm that the proposed new initial value of the alternating-directions implicit (ADI) can lead to a significant reduction in the total number of ADI steps, while also showing a 17% and 8x speed-up over the zero initial value for the two equation types, respectively.

翻译：基于交换算子分裂，我们推导了交替方向隐式（ADI）方法，并将结果应用于具有低秩常数项和近似解的连续时间代数Lyapunov方程。以往，低秩ADI（LR-ADI）必须从全零初值开始迭代。我们的方法保留了LR-ADI方法中已知的高效低秩增量与残差迭代格式，并将其推广至任意低秩初值情形。我们进一步将Lyapunov方程LR-ADI的若干已知性质推广到更广泛的算法类别或问题类型中。通过两个典型调用LR-ADI的外层迭代，我们研究了任意初值的性能表现：首先，采用牛顿法求解代数Riccati方程；其次，采用一阶Rosenbrock方法求解微分Riccati方程。数值实验证实，所提出的交替方向隐式（ADI）新初值策略能显著减少ADI总迭代步数，同时在两类方程求解中分别实现了相较于零初值17%的加速和8倍的提速。