Improved methods and starting values to solve the matrix equations iteratively

The two matrix iterations are known to converge linearly to a positive definite solution of the matrix equations , respectively, for known choices of and under certain restrictions on . The convergence for previously suggested starting matrices is generally very slow. This paper explores different initial choices of in both iterations that depend on the extreme singular values of 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.