A new class of complex nonsymmetric algebraic Riccati equations

TL;DR

This paper introduces a generalized class of complex nonsymmetric algebraic Riccati equations, establishes the existence and uniqueness of extremal solutions, and analyzes the convergence of classical algorithms for solving them.

Contribution

It proposes a new parameterized comparison matrix, extends the class of NAREs, and provides convergence analysis and strategies for classical solution algorithms.

Findings

Newton's method is quadratically convergent.

Fixed-point iterative method is linearly convergent.

Doubling algorithms are quadratically convergent with proper parameters.

Abstract

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.