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1 Introduction Inthispaper,weconsidertheunconstrained optimizationproblem minf(x)(1.1) wheref(x):Rn→Ristwicecontinuouslydifferen tiable,andasolutionx totheproblemexists.Qua si Newtonmethodsforsolvingproblem(1.1)are widelyapplied.Startingfromaninitialpoint