Convergence Analysis for Several Iterative Methods of Coupled Nonsymmetric Algebraic Riccati Equations

In this paper, we consider coupled nonsymmetric algebraic Riccati equations (CNAREs) arising in jump linear quadratic differential system. Firstly, we show that the minimal positive matrix m-tuple solutions of the CNAREs exist under some specific assumptions. Secondly, we review the basic fixed-point iterative method for solving the CNAREs, and prove that the fixed-point iteration is linearly or sublinearly convergent. Thirdly, we propose Newton’s method to solve the CNAREs, and show that Newton’s iteration is quadratically or linearly convergent. We figure out that our convergence analysis about the fixed-point iteration corrects and refines the analysis given in a recent paper (Zhang et al. in J Appl Math Comput 68:4119–4133, 2022), and our proposed Newton’s method is definitely the standard Newton’s method which differs from that proposed in a recent paper (Liu et al. in Comput Appl Math 39(2):1–17, 2020). Besides, we also devise a mixed algorithm by executing several steps of fixed-point iterations and then switching to Newton’s iterations and give its convergence results. Finally, we also give some examples to demonstrate the validity and applicability of the proposed methods.