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We consider the new family of two step Runge–Kutta– Nyström methods for the numerical integration of y″=f(x,y), which provide approximation for the solution and its first derivative at the step point, and depend on the stage values at two consecutive step points. We derive the conditions to obtain methods within this family, which integrate algebraic polynomials exactly, describe a constructive technique and analyze the order of the resulting method.

A General Family of Two Step Runge-Kutta-Nyström Methods for y ″ = f(x,y) Based on Algebraic Polynomials