A new class of complex nonsymmetric algebraic Riccati equations
This work extends the theoretical framework of algebraic Riccati equations to a broader class of complex matrices, which is incremental for researchers in numerical linear algebra and control theory.
The authors propose a new class of complex nonsymmetric algebraic Riccati equations (NAREs) by generalizing the definition of comparison matrices, and prove existence and uniqueness of extremal solutions. They demonstrate quadratic convergence for Newton's method and doubling algorithms, and linear convergence for fixed-point iterative methods, with numerical experiments validating parameter strategies.
In this paper, we first propose a new parameterized definition of comparison matrix of a given complex matrix, which generalizes the definition proposed by \cite {Axe1}. Based on this, we propose a new class of complex nonsymmetric algebraic Riccati equations (NAREs) which extends the class of nonsymmetric algebraic Riccati equations proposed by \cite {Axe1}. We also generalize the definition of the extremal solution of an NARE and show that the extremal solution of an NARE exists and is unique. Some classical algorithms can be applied to search for the extremal solution of an NARE, including Newton's method, some fixed-point iterative methods and doubling algorithms. Besides, we show that Newton's method is quadratically convergent and the fixed-point iterative method is linearly convergent. We also give some concrete strategies for choosing suitable parameters such that the doubling algorithms can be used to deliver the extremal solutions, and show that the two doubling algorithms with suitable parameters are quadratically convergent. Numerical experiments show that our strategies for parameters are effective.