SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 59-83, March 2024.
Abstract. Structure-preserving doubling algorithms (SDAs) are efficient algorithms for solving Riccati-type matrix equations. However, breakdowns may occur in SDAs. To remedy this drawback, in this paper, we first introduce [math]-symplectic forms ([math]-SFs), consisting of symplectic matrix pairs with a Hermitian parametric matrix [math]. Based on [math]-SFs, we develop modified SDAs (MSDAs) for solving the associated Riccati-type equations. MSDAs generate sequences of symplectic matrix pairs in [math]-SFs and prevent breakdowns by employing a reasonably selected Hermitian matrix [math]. In practical implementations, we show that the Hermitian matrix [math] in MSDAs can be chosen as a real diagonal matrix that can reduce the computational complexity. The numerical results demonstrate a significant improvement in the accuracy of the solutions by MSDAs.

中文翻译：

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避免代数 Riccati 型矩阵方程崩溃的结构保持加倍算法

《SIAM 矩阵分析与应用杂志》，第 45 卷，第 1 期，第 59-83 页，2024 年 3 月。
摘要。结构保持加倍算法 (SDAs) 是求解 Riccati 型矩阵方程的有效算法。然而，SDAs 可能会发生故障。为了弥补这个缺点，在本文中，我们首先引入[math]-辛形式（[math]-SFs），由带有埃尔米特参数矩阵[math]的辛矩阵对组成。基于数学-SF，我们开发了改进的 SDA（MSDA）来求解相关的 Riccati 型方程。MSDA 在 [math]-SF 中生成辛矩阵对序列，并通过采用合理选择的埃尔米特矩阵 [math] 来防止崩溃。在实际实现中，我们表明可以选择 MSDA 中的 Hermitian 矩阵 [math] 作为实数对角矩阵，从而降低计算复杂度。数值结果表明 MSDA 解决方案的准确性有了显着提高。