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

The two matrix iterations $X_{k+1}=I\mp A^*X_k^{-1}A$ are known to converge linearly to a positive definite solution of the matrix equations $X\pm A^*X^{-1}A=I$, respectively, for known choices of $X_0$ and under certain restrictions on $A$. The convergence for previously suggested starting matrices $X_0$ is generally very slow. This paper explores different initial choices of $X_0$ 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.