On Smith-type iterative algorithms for the Stein matrix equation

Abstract

This note studies the iterative solution to the Stein matrix equation. Firstly, it is shown that the recently developed Smith$\left(l\right)$ 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 $r$-Smith iteration that includes the well-known ordinary Smith accelerative iteration as a special case, we have shown that the $r$-Smith accelerative iteration requires less computation than the Smith iteration and the Smith$\left(l\right)$ iteration, and the ordinary Smith accelerative iteration requires the least computations comparing with other Smith-type iterations.