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In this paper we will study a low algebraic order four-step method requiring this specific method to have vanished the phase-lag and its first and second derivatives. For this specific method we will give the constant value of the parameters in the right hand side part of the method. We will investigate the influence of the elimination of the phase-lag and its first and second derivatives on the efficiency of this method. More specifically we will study the local truncation error of the new method and we will compare it with other methods in the literature (comparative local truncation error analysis). We will also investigate the stability (interval of periodicity) of this method using scalar test equation with frequency different than the frequency of the scalar test equation used for phase-lag analysis (stability analysis). Finally, the new produced method will be applied on the resonance problem of the Schr\odinger equation in order to examine its efficiency. We will prove that this kind of methods are effective for the approximate solution of the Schr\"odinger equation and related periodic initial-value or boundary-value problems."

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