RecentlyY.Saadproposedaflexibleinner-outerpreconditionedGMRESalgorithmfornonsymmetriclinearsystems[4].Followingtheirideas,wesuggestanadaptivepreconditionedCGSmethod,calledCGS/GMRES(k),inwhichthepreconditionerisconstructedintheiterationstepofCGS,byseveralstepsofGMRES(k).Numericalexperimentsshowthattheresidualoftheouteriterationdecreasesrapidly.WealsofoundtheinterestingresidualbehaviourofGMRESfortheskewsymmetriclinearsystemAx=b,whichgivesaconvergenceresultforrestartedGMRES(k).Forconvenience,wediscussrealsystems.
