On a new iterative method for solving linear systems and comparison results

In Ujević [A new iterative method for solving linear systems, Appl. Math. Comput. 179 (2006) 725–730], the author obtained a new iterative method for solving linear systems, which can be considered as a modification of the Gauss–Seidel method. In this paper, we show that this is a special case from a point of view of projection techniques. And a different approach is established, which is both theoretically and numerically proven to be better than (at least the same as) Ujević's. As the presented numerical examples show, in most cases, the convergence rate is more than one and a half that of Ujević.