Improved methods and starting values to solve the matrix equations X +/- A*X(-1)A=I iteratively

摘要

The two matrix iterations Xk+1 = I -/+ A*X(k)(-1)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.