A class of complex nonsymmetric algebraic Riccati equations associated with H-matrix

Abstract

In this paper, based on the study of fluid flow models modulated by a Markov chain, we propose a new class of complex nonsymmetric algebraic Riccati equations (NAREs) associated with H-matrix. Under the assumption that the diagonal elements of the associated H-matrix are on a same ray in the complex plane, we generalize the definition of the extremal solution of an H-matrix algebraic Riccati equation (HARE) whose associated H-matrix has positive or negative diagonal elements, and show that the extremal solution exists and is unique. Besides, we show that the basic fixed-point iterative methods are linearly convergent and the Newton’s method is quadratically convergent when they are applied to search for the extremal solutions of the HAREs. We also give out criteria for choosing suitable parameters such that the three existing doubling algorithms can deliver the extremal solutions of the HAREs, and show that the doubling algorithms are quadratically convergent. More importantly, we also propose a preprocessing technique to transform an HARE under a certain assumption into a new HARE under the aforementioned assumption. Numerical experiments show that our methods and preprocessing technique are effective.