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A Runge-Kutta type eighth algebraic order two-step method with phase-lag and its first, second and third order derivatives equal to zero is produced in this paper. We will also investigate how the above described elimination of the phase-lag and its derivatives effects on the efficiency of the method. More specifically we will study the following: (1) the production of the method, (2) the local truncation error of the new obtained method and a comparative local truncation error analysis using other similar methods of the literature, (3) the interval of periodicity i.e the stability of the developed method using frequency for the scalar test equation for the stability analysis different than the frequency used in the scalar test equation for phase-lag analysis and (4) the effectiveness of the new obtained method applying it on the resonance problem of the radial Schr¨odinger equation. Based on the last study we will show the efficiency of new method.

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