An effective algorithm for differential equations of motion

The paper presents eighth-order method for solution of differential equation. Legendre's nodes are used as argument of derivatives computing. That is why there are only four points per step for derivatives computing. This results decrease numerical noise. Adams type predictor-corrector is used to obtain initial values for single iteration of implicit Runge–Kutta algorithm, so the derivatives are computed twice for the each step. All quadrature coefficients which integrator uses are calculated beforehand (for uniform step, increasing step, decreasing step) to obtain minimum possible volume of computation.