On Smith-type iterative algorithms for the Stein matrix equation

This note studies the iterative solution to the Stein matrix equation. Firstly, it is shown that the recently developed Smith(l)(l) iteration converges to the exact solution for arbitrary initial condition whereas a special initial condition is required in the literature. Secondly, by presenting a new accelerative Smith iteration named the rr-Smith iteration that includes the well-known ordinary Smith accelerative iteration as a special case, we have shown that the rr-Smith accelerative iteration requires less computation than the Smith iteration and the Smith(l)(l) iteration, and the ordinary Smith accelerative iteration requires the least computations comparing with other Smith-type iterations.