Fast and Inversion-Free Iterative Algorithms for Discrete Periodic Riccati Equations

This paper presents two novel iterative algorithms for solving discrete periodic Riccati equations (DPRE) arising in linear periodic systems. The first algorithm employs a full latest-estimate strategy that maximizes the utilization of freshly computed solutions within each iteration, significantly accelerating convergence compared to existing methods. The second algorithm eliminates matrix inversion operations entirely through an equivalent transformation and integrates the latest-estimate strategy for further efficiency. Both algorithms are proven to converge monotonically to the unique positive definite solution. Numerical simulations demonstrate that the first algorithm achieves faster convergence, while the second algorithm shows superior computational efficiency in high-dimensional systems by avoiding costly matrix inversions.