A note on the preconditioned Gauss–Seidel (GS) method for linear systems

In this note recent comparison results for preconditioned Gauss–Seidel (GS) methods are discussed. A new strict comparison result between two different preconditioned GS methods is given, some errors in a recent article by Niki et al. (J. Comput. Appl. Math. 164–165 (2004) 587) are pointed out and a new proof for the corresponding results in Niki et al. is presented.