In this work we develop a low-rank algorithm for the computation of low-rank approximations to the large-scale Lyapunov operator $\varphi$-functions. Such computations constitute the important ingredient to the implementation of matrix-valued exponential integrators for large-scale stiff matrix differential equations where the (approximate) solutions are of low rank. We evaluate the approximate solutions of $LDL^T$-type based on a scaling and recursive procedure. The key parameters of the method are determined using a quasi-backward error bound combined with the computational cost. Numerical results demonstrate that our method can be used as a basis of matrix-valued exponential integrators for solving large-scale differential Lyapunov equations and differential Riccati equations.
翻译:在这项工作中,我们为计算大型Lyapunov操作员的低位近似值制定了一种低级算法。这种计算是实施大型硬基体差异方程式的矩阵估值指数集成器的重要因素,因为(近似)解决方案处于低位。我们根据一个缩放和循环程序评估了$LDL ⁇ T$类型的近似解决方案。该方法的关键参数是使用一个准背向错误来确定的,与计算成本相结合。数字结果表明,我们的方法可以用作矩阵估值指数集成器的基础,用以解决大型的Lyapunov方程式和差异里卡提方程式。