The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

In this paper we propose a new algorithm for solving large-scale algebraic Riccati equations with low-rank structure. The algorithm is based on the found Toeplitz-structured closed form of the stabilizing solution and the fast Fourier transform. It works without unnecessary assumptions, shift selection trategies, or matrix calculations of the cubic order with respect to the problem scale. Numerical examples are given to illustrate its features. Besides, we show that it is theoretically equivalent to several algorithms existing in the literature in the sense that they all produce the same sequence under the same parameter setting.