Transforming algebraic Riccati equations into unilateral quadratic matrix equations (Q711584)

Given \(A\in\mathbb{R}^{m\times m}\), \(B\in\mathbb{R}^{m\times n}\), \(C\in\mathbb{R}^{n\times m}\) and \(D\in\mathbb{R}^{n\times n}\), the authors reduce the nonsymmetric algebraic Riccati equation (ARE), \(XCX-AX-XD+B=0\), where \(X\in\mathbb{R}^{m\times n}\) is the unknown, to a unitaral quadratic matrix equation (UQME), \(\mathcal{P}\mathcal{X}^2+\mathcal{Q}\mathcal{X}+\mathcal{R}=0\), by means of a UL-based transformation and an eigenvalue transformation. Here, the matrices have size \(m+n\) but have a strong structure. These transformations lead to a unified framework for the structure-preserving doubling algorithm (SDA) of \textit{B. D. O. Anderson} [Int. J. Control 28, 295-306 (1978; Zbl 0385.49017)], and \textit{X.-X. Guo} et al. [Numer. Math. 103, No. 3, 393--412 (2006; Zbl 1097.65055)], and the cyclic reduction (CR) of \textit{R. W. Hockney} [J. Assoc. Comput. Mach. 12, 95--113 (1965; Zbl 0139.10902)]. This fact enables one to deduce the convergence properties of SDA directly from CR when SDA is applied to UQME. Moreover, this framework allows one to design new algorithms for the effective solution of ARE. In particular, by complementing the eigenvalue transformation with the shrink-and-shift technique of Ramaswami, one obtains a new algorithm having the same cost per iteration as SDA but relying on a simpler initialization.