The intrinsic Toeplitz structure and its applications in algebra Riccati equations

In this paper we propose a new algorithm for solving large-scale algebraic Riccati equations with low-rank structure, which is based on the found elegant closed form of the stabilizing solution that involves an intrinsic Toeplitz structure and the fast Fourier transform used to accelerate the multiplication of a Toeplitz matrix and vectors. The algorithm 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.