Improved methods and starting values to solve the matrix equations $X\pm A^*X^{-1}A=I$ iteratively

Abstract

The two matrix iterations X k+1 = I ∓ A * X k-1A are known to converge linearly to a positive definite solution of the matrix equations X ± A*X -1A = I, respectively, for known choices of X o and under certain restrictions on A. The convergence for previously suggested starting matrices X o is generally very slow. This paper explores different initial choices of X o in both iterations that depend on the extreme singular values of A and lead to much more rapid convergence. Further, the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use Newton's method in part.