WediscussavariantofrestartedGMRESmethodthatallowschangesoftherestartingvectorateachcycleofiterations.Themeritofthevariantisthatpreviouslygeneratedinformationcanbeutilizedtoselectanewstartingvector,suchthattheoccurrenceofstagnationbemitigatedortheconvergencebeaccelerated.ThemoreappealingutilizationofthenewmethodisinconjunctionwithaharmonicRitzvectorasthestartingvector,whichisdiscussedindetail.Numericalexperimentsarecarriedouttodemonstratethattheproposedprocedurecaneffectivelymitigatetheoccurrenceofstagnationduetothepresenceofsmalleigenvaluesinmodulus.