Iterative algorithms for X+ATX−1A=I by using the hierarchical identification principle

By using the hierarchical identification principle and introducing the convergence factor and the iterative matrix, a family of inversion-free iterative algorithms is proposed for solving nonlinear matrix equations X+ATX−1A=I. The convergence is proved and the convergence speed is analyzed. The suggested iterative algorithm includes some previous algorithms as its special cases. Two numerical examples are given to illustrate the proposed algorithms. Theoretical analysis and numerical examples show that for appropriate convergence factors, the proposed algorithms are more efficient than the existing inversion-free iterative algorithms.