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.

中文翻译：

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一类新的复数非对称代数Riccati方程，其ω-比较矩阵是不可约奇异M-矩阵

摘要 在本文中，我们提出并讨论了一类新的复杂非对称代数 Riccati 方程 (NARE)，其四个系数矩阵形成一个矩阵，其 ω 比较矩阵是不可约奇异 M 矩阵。我们还证明了 NARE 的极值解在非临界情况下唯一存在，在临界情况下也存在。还显示了解决方案的一些良好特性。此外，还应用了一些经典的数值方法，包括Schur方法、Newton方法、定点迭代方法和倍增算法，来求解NARE，并详细给出了这些方法的收敛性分析。对于加倍算法，我们也给出了具体的参数选择策略。数值结果表明我们的方法对于解决 NARE 是有效的。