A new class of complex nonsymmetric algebraic Riccati equations with its ω-comparison matrix being an irreducible singular M-matrix

ABSTRACT

In this paper, we propose and discuss a new class of complex nonsymmetric algebraic Riccati equations (NAREs) whose four coefficient matrices form a matrix with its ω-comparison matrix being an irreducible singular M-matrix. We also prove that the extremal solutions of the NAREs exist uniquely in the noncritical case and exist in the critical case. Some good properties of the solutions are also shown. Besides, some classical numerical methods, including the Schur methods, Newton's method, the fixed-point iterative methods and the doubling algorithms, are also applied to solve the NAREs, and the convergence analysis of these methods are given in details. For the doubling algorithms, we also give out the concrete parameter selection strategies. The numerical results show that our methods are efficient for solving the NAREs.

Keywords:

• Complex nonsymmetric algebraic Riccati equation
• extremal solution
• classical numerical method
• doubling algorithm
• parameter selection strategy

2010 AMS Subject Classifications:

• 65F15
• 65F18
• 65F30

Acknowledgments

We also deeply appreciate the anonymous referees for their helpful suggestive comments and suggestions, which have helped us improve the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by National Natural Science Foundation of China (11671318).