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The two matrix iterations X k+1 = I A*X -1 k A are known to converge linearly to a positive definite solution of the matrix equations X ± 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.

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