论文标题
Riccati adi：存在，独特性和新的迭代方法
Riccati ADI: Existence, uniqueness and new iterative methods
论文作者
论文摘要
考虑了大规模代数riccati方程的近似解。我们对近似解决方案感兴趣，该解决方案产生了特定小等级的Riccati残留矩阵。假定可以用矩形矩阵$ z $和小的二次矩阵$ y $ $ y $编写这种近似解决方案。我们建议选择$ z $，以便其列涵盖某个理性的Krylov子空间。存在这种近似解决方案并确定唯一的条件。结果表明，所提出的方法可以解释为倾斜投影方法。有两个新的迭代程序，具有有效的解决方案和残留因子的更新。通过我们的方法，可以处理复杂系统矩阵，提供实现并引入并行化。
The approximate solution of large-scale algebraic Riccati equations is considered. We are interested in approximate solutions which yield a Riccati residual matrix of a particular small rank. It is assumed that such approximate solutions can be written in factored form $ZYZ^*$ with a rectangular matrix $Z$ and a small quadratic matrix $Y$. We propose to choose $Z$ such that its columns span a certain rational Krylov subspace. Conditions under which such an approximate solution exists and is unique are determined. It is shown that the proposed method can be interpreted as an oblique projection method. Two new iterative procedures with efficient updates of the solution and the residual factor are derived. With our approach complex system matrices can be handled, realification is provided and parallelization is introduced.
