Structure-Preserving Doubling Algorithms That Avoid Breakdowns for Algebraic Riccati-Type Matrix Equations

Abstract

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.